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<title>Differentiationsklasse</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Differentiationsklasse</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Die Differentiationsklasse<a href="#cite_note-1">[1]</a> ist ein Begriff aus der <a href="Mathematik" title="Mathematik">Mathematik</a>, insbesondere aus dem Teilgebiet der <a href="Analysis" title="Analysis">Analysis</a>. Sie ist ein <a href="Funktionenraum" title="Funktionenraum">Funktionenraum</a> und umfasst alle Funktionen, die mindestens <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-mal <a href="Stetig_differenzierbare_Funktion" class="mw-redirect" title="Stetig differenzierbare Funktion">stetig differenzierbar</a> sind, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> eine <a href="Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürliche Zahl</a> ist. Notiert wird die Differentiationsklasse meist mittels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle C^{k}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} \cup \{0\}}">

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<annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} \cup \{0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d22167c127c41e7d58f35200ed1d2a7357889201.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.8ex; height:2.843ex;" alt="{\displaystyle k\in \mathbb {N} \cup \{0\}}" loading="lazy"> eine Zahl und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\subset \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle D\subset \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1798221e1270f83ece4ead9abf0bf0c06c041e23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.701ex; height:2.176ex;" alt="{\displaystyle D\subset \mathbb {R} }" loading="lazy"> eine nichtleere, <a href="Offene_Menge" title="Offene Menge">offene</a> <a href="Teilmenge" title="Teilmenge">Teilmenge</a> der <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a>. Eine <a href="Stetige_Funktion" title="Stetige Funktion">stetige Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon D\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle f\colon D\to \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/229e3a78dd4be03159691c3d4c84a9095784d26e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.529ex; height:2.509ex;" alt="{\displaystyle f\colon D\to \mathbb {R} }" loading="lazy"> gehört dann zur Differentiationsklasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}(D)}">

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<annotation encoding="application/x-tex">{\displaystyle C^{k}(D)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f326dab3fb9dc402145ae09d3929ed89cfca69b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.62ex; height:3.176ex;" alt="{\displaystyle C^{k}(D)}" loading="lazy"> beziehungsweise genauer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}(D,\mathbb {R} )}">

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<annotation encoding="application/x-tex">{\displaystyle C^{k}(D,\mathbb {R} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50752d74f2276d3d68e164b81e9c148a6a8460cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.332ex; height:3.176ex;" alt="{\displaystyle C^{k}(D,\mathbb {R} )}" loading="lazy">, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> auf ganz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> mindestens <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-mal <a href="Stetig_differenzierbare_Funktion" class="mw-redirect" title="Stetig differenzierbare Funktion">stetig differenzierbar</a> ist.<a href="#cite_note-Koenigsberger155-2">[2]</a>

Entsprechend der Definition wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(D):=C^{0}(D)}">

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<annotation encoding="application/x-tex">{\displaystyle C(D):=C^{0}(D)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/756acfe5286c338bfdd969c9ff595352a5f88409.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.831ex; height:3.176ex;" alt="{\displaystyle C(D):=C^{0}(D)}" loading="lazy"> die Klasse der stetigen Funktionen und mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(D)}">

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<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(D)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9e622ef8b8c7be192cd6075143b7f919ba77e98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.407ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(D)}" loading="lazy"> die Differentiationsklasse der <a href="Glatte_Funktion" title="Glatte Funktion">beliebig oft differenzierbaren Funktionen</a> bezeichnet.<a href="#cite_note-Koenigsberger155-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Verallgemeinerungen">Verallgemeinerungen</h2></div>

Die Klasse der <a href="Analytische_Funktion" title="Analytische Funktion">analytischen Funktionen</a> wird manchmal in Analogie zu obiger Definition mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\omega }(D)}">

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<annotation encoding="application/x-tex">{\displaystyle C^{\omega }(D)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/581453c42d313b70c49fe534e1104873178b6d26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.786ex; height:2.843ex;" alt="{\displaystyle C^{\omega }(D)}" loading="lazy"> bezeichnet.

Für stetige Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon {\tilde {D}}\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}">

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<annotation encoding="application/x-tex">{\displaystyle g\colon {\tilde {D}}\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c13c6ff1b6f923b4eec9f456ebd65a3ddf84d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.036ex; height:3.009ex;" alt="{\displaystyle g\colon {\tilde {D}}\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}" loading="lazy"> im mehrdimensionalen <a href="Euklidischer_Vektorraum" class="mw-redirect" title="Euklidischer Vektorraum">euklidischen Vektorraum</a> wird die Definition analog übernommen. Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> gehört also zur Differentiationsklasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}({\tilde {D}},\mathbb {R} ^{m})}">

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<annotation encoding="application/x-tex">{\displaystyle C^{k}({\tilde {D}},\mathbb {R} ^{m})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e4cdbdf1907fdfe802f882b830dbc52c5d0ee40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.007ex; height:3.176ex;" alt="{\displaystyle C^{k}({\tilde {D}},\mathbb {R} ^{m})}" loading="lazy">, wenn sie auf ganz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {D}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {D}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/151fff560f471c0ce3c382dc71b908567e09ac6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle {\tilde {D}}}" loading="lazy"> mindestens <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-mal <a href="Stetig_differenzierbare_Funktion#Reellwertige_Funktionen_mehrerer_Variablen" class="mw-redirect" title="Stetig differenzierbare Funktion">stetig differenzierbar</a> ist.<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a>

Wenn sich die Anzahl der möglichen Differentiationen (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k,l,..}">

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<annotation encoding="application/x-tex">{\displaystyle k,l,..}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bef771651777c0b328afb4adda02647afebbbd83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.653ex; height:2.509ex;" alt="{\displaystyle k,l,..}" loading="lazy">) bei mehrdimensionalen Funktionen zwischen den einzelnen Variablen unterscheidet, so kann dem in einer Verallgemeinerung der obigen Notation Rechnung getragen werden: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k,l,..}(D).}">

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<annotation encoding="application/x-tex">{\displaystyle C^{k,l,..}(D).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/222045b06b7bdf704987d3133bb51dea83bfab0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.587ex; height:3.176ex;" alt="{\displaystyle C^{k,l,..}(D).}" loading="lazy"><a href="#cite_note-5">[5]</a>

Auch für Funktionen zwischen <a href="Differenzierbare_Mannigfaltigkeit" title="Differenzierbare Mannigfaltigkeit">differenzierbaren Mannigfaltigkeiten</a> werden die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle C^{k}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy">-Differentiationsklassen analog definiert.

<div class="mw-heading mw-heading2"><h2 id="Teilmengenrelation">Teilmengenrelation</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\subset \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle D\subset \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34223d8343154b3007fb4e9fd88285a37e4cd53f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.919ex; height:2.343ex;" alt="{\displaystyle D\subset \mathbb {R} ^{n}}" loading="lazy"> eine offene Teilmenge, dann gilt

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<annotation encoding="application/x-tex">{\displaystyle C^{\omega }(D)\subsetneq C^{\infty }(D)\subsetneq \dotsb \subsetneq C^{k}(D)\subsetneq \dotsb \subsetneq C^{1}(D)\subsetneq C^{0}(D)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6fa3ec487df2cb032b13f89f3ed674d526ace1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.021ex; height:3.176ex;" alt="{\displaystyle C^{\omega }(D)\subsetneq C^{\infty }(D)\subsetneq \dotsb \subsetneq C^{k}(D)\subsetneq \dotsb \subsetneq C^{1}(D)\subsetneq C^{0}(D)}" loading="lazy">.</dd></dl>

Je höher also der Index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> der Differentiationsklasse ist, desto weniger Funktionen umfasst sie.

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<ul><li>Die <a href="Exponentialfunktion" title="Exponentialfunktion">Exponentialfunktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp \colon \mathbb {R} \rightarrow \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \exp \colon \mathbb {R} \rightarrow \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ce9e767d8392ff55678f23a48c979ca38c41010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.557ex; height:2.509ex;" alt="{\displaystyle \exp \colon \mathbb {R} \rightarrow \mathbb {R} }" loading="lazy"> ist analytisch und gehört somit zur Klasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\omega }(\mathbb {R} )}">

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<annotation encoding="application/x-tex">{\displaystyle C^{\omega }(\mathbb {R} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41b09f7854c08a7e4103a4e2fdd65414e3c6d06d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.54ex; height:2.843ex;" alt="{\displaystyle C^{\omega }(\mathbb {R} )}" loading="lazy">.</li>

<li>Die <a href="Betragsfunktion" title="Betragsfunktion">Betragsfunktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |\colon \mathbb {R} \to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle |\cdot |\colon \mathbb {R} \to \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60a765868bd0ec6aa8f11275272e35f22724e215.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.977ex; height:2.843ex;" alt="{\displaystyle |\cdot |\colon \mathbb {R} \to \mathbb {R} }" loading="lazy"> ist stetig, aber nicht differenzierbar. Sie gehört also zur Klasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{0}(\mathbb {R} )}">

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<annotation encoding="application/x-tex">{\displaystyle C^{0}(\mathbb {R} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6cd1d95cbad44565585e12b18114eecd255288e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.339ex; height:3.176ex;" alt="{\displaystyle C^{0}(\mathbb {R} )}" loading="lazy">, aber nicht zur Klasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}(\mathbb {R} )}">

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<annotation encoding="application/x-tex">{\displaystyle C^{1}(\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9aafd3a446ae8e5a01fcf831b4b07a70ff9004aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.339ex; height:3.176ex;" alt="{\displaystyle C^{1}(\mathbb {R} )}" loading="lazy">.</li>

<li>Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1cacd5f7bbe1027cc75fbe2fbd9cb5e79485302.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto |x|^{3}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\mapsto |x|^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c5b7563aef9cb7c5479ad93ab2bd79435ddbcf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.621ex; height:3.343ex;" alt="{\displaystyle x\mapsto |x|^{3}}" loading="lazy"> ist zweimal stetig differenzierbar, aber nicht dreimal. Es gilt also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C^{2}(\mathbb {R} )\setminus C^{3}(\mathbb {R} )}">

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in C^{2}(\mathbb {R} )\setminus C^{3}(\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3122cf08b582672d048a711e2855e2ce5c0b4ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.993ex; height:3.176ex;" alt="{\displaystyle f\in C^{2}(\mathbb {R} )\setminus C^{3}(\mathbb {R} )}" loading="lazy">.</li>

<li>Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}\colon \mathbb {R} \to \mathbb {R} }">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{n}\colon \mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32c2f924f3241272652ec2410c185a92f7236a28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.362ex; height:2.509ex;" alt="{\displaystyle f_{n}\colon \mathbb {R} \to \mathbb {R} }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x^{n}|x|}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{n}|x|}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e883eb5ed2af9d5a481ec151cfb1d9f9932253b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.115ex; height:2.843ex;" alt="{\displaystyle x\mapsto x^{n}|x|}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-mal stetig differenzierbar, aber nicht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (n+1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b30a29cfd35628469f9dbffea4804f5b422f3037.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n+1)}" loading="lazy">-mal für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy">. Es gilt also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}\in C^{n}(\mathbb {R} )\setminus C^{n+1}(\mathbb {R} )}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{n}\in C^{n}(\mathbb {R} )\setminus C^{n+1}(\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd041a286fe51326ad7bf341d9f2c90aa29afba0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.501ex; height:3.176ex;" alt="{\displaystyle f_{n}\in C^{n}(\mathbb {R} )\setminus C^{n+1}(\mathbb {R} )}" loading="lazy">.</li>

<li>Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon \mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\colon \mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/926a8888ada479f404e300131e232ca43f4c867b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.12ex; height:2.509ex;" alt="{\displaystyle g\colon \mathbb {R} \to \mathbb {R} }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=\exp \left(-{\tfrac {1}{x^{2}}}\right)}">

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)=\exp \left(-{\tfrac {1}{x^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bca8d7f1f687f7b8732c07333fe9ee8f761f1e59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.097ex; height:4.843ex;" alt="{\displaystyle g(x)=\exp \left(-{\tfrac {1}{x^{2}}}\right)}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35a455db7b2aab1b0e72ccbc7385e4424e2372e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(0)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(0)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/679b33e306b0721551440784cbc6b73cc13c77d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.349ex; height:2.843ex;" alt="{\displaystyle g(0)=0}" loading="lazy"> ist beliebig oft differenzierbar und gehört somit zur Klasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(\mathbb {R} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c913ced284ad05bd42c8f004a3a83dee6323cb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.161ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(\mathbb {R} )}" loading="lazy">, aber sie ist nicht analytisch.</li>

<li>Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\colon \mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\colon \mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aad637bebb261539130e516d148b93b069349c7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.343ex; height:2.176ex;" alt="{\displaystyle h\colon \mathbb {R} \to \mathbb {R} }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)=x^{2}\sin(1/x)}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(x)=x^{2}\sin(1/x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d8f965f50100c5bb9779d50ec59188c6fe536f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.667ex; height:3.176ex;" alt="{\displaystyle h(x)=x^{2}\sin(1/x)}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35a455db7b2aab1b0e72ccbc7385e4424e2372e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(0)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(0)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b76788b935157a38df72362e3c4eb85323f2c7bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.572ex; height:2.843ex;" alt="{\displaystyle h(0)=0}" loading="lazy"> ist überall differenzierbar, aber die Ableitungsfunktion ist an der Stelle Null nicht stetig. Somit gehört die Funktion nicht zur Klasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}(\mathbb {R} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{1}(\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9aafd3a446ae8e5a01fcf831b4b07a70ff9004aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.339ex; height:3.176ex;" alt="{\displaystyle C^{1}(\mathbb {R} )}" loading="lazy">, sondern nur zur Klasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{0}(\mathbb {R} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{0}(\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6cd1d95cbad44565585e12b18114eecd255288e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.339ex; height:3.176ex;" alt="{\displaystyle C^{0}(\mathbb {R} )}" loading="lazy">.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Genügend_glatt">Genügend glatt</h2></div>

Im Zusammenhang mit der Differenzierbarkeit wird manchmal davon gesprochen, dass eine Funktion genügend glatt sei. Dies bedeutet, dass im jeweiligen Kontext genügend oft differenzierbar ist, man sich also gewissermaßen keine zusätzlichen Gedanken um die Differenzierbarkeit machen muss.<a href="#cite_note-6">[6]</a> Der Begriff leitet sich aus der Bezeichnung <a href="Glatte_Funktion" title="Glatte Funktion">glatte Funktion</a> für eine beliebig oft differenzierbare Funktion ab.

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Rolf Walter: Einführung in die Analysis. Band 3. Walter de Gruyter, Berlin u. a. 2009, ISBN 978-3-11-020960-0, S. 59, 147ff.

</li>

<li id="cite_note-Koenigsberger155-2">↑ <a href="#cite_ref-Koenigsberger155_2-0">a</a> <a href="#cite_ref-Koenigsberger155_2-1">b</a> <a href="Konrad_K%C3%B6nigsberger" title="Konrad Königsberger">Konrad Königsberger</a>: Analysis 1. Springer-Verlag, Berlin u. a., 2004, ISBN 3-540-41282-4, S. 155.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> <a href="Konrad_K%C3%B6nigsberger" title="Konrad Königsberger">Konrad Königsberger</a>: Analysis 2. Springer-Verlag, Berlin/Heidelberg, 2000, ISBN 3-540-43580-8, S. 62.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Rolf Walter: <cite style="font-style:italic">Einführung in die Analysis 2</cite>. de Gruyter, 2007, ISBN 978-3-11-019540-8, S. 64, 448.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Differentiationsklasse&rft.au=Rolf+Walter&rft.btitle=Einf%C3%BChrung+in+die+Analysis+2&rft.date=2007&rft.genre=book&rft.isbn=9783110195408&rft.pages=64%2C+448&rft.pub=de+Gruyter" style="display:none">

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Martin Keller-Ressel: <a rel="nofollow" class="external text" href="https://tu-dresden.de/mn/math/stochastik/mkeller/ressourcen/dateien/stochana_skript.pdf">Stochastische Analysis.</a> (PDF; 838 kB) In: tu-dresden.de. <a href="Technische_Universit%C3%A4t_Dresden" title="Technische Universität Dresden">TU Dresden</a>, Fakultät Mathematik, 23. Mai 2015, S. 46, abgerufen am 23. Mai 2024.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3ADifferentiationsklasse&rft.title=Stochastische+Analysis&rft.description=Stochastische+Analysis&rft.identifier=https%3A%2F%2Ftu-dresden.de%2Fmn%2Fmath%2Fstochastik%2Fmkeller%2Fressourcen%2Fdateien%2Fstochana_skript.pdf&rft.creator=Martin+Keller-Ressel&rft.publisher=%5B%5BTechnische+Universit%C3%A4t+Dresden%7CTU+Dresden%5D%5D%2C+Fakult%C3%A4t+Mathematik&rft.date=2015-05-23&rft.language=de">

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> Dirk Langemann, Cordula Reisch: <cite style="font-style:italic">So einfach ist Mathematik – Partielle Differenzialgleichungen für Anwender</cite>. 1. Auflage. Springer Spektrum, Berlin / Heidelberg 2018, ISBN 978-3-662-57501-7, S. 101.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Differentiationsklasse&rft.au=Dirk+Langemann%2C+Cordula+Reisch&rft.btitle=So+einfach+ist+Mathematik+-+Partielle+Differenzialgleichungen+f%C3%BCr+Anwender&rft.date=2018-12-10&rft.edition=1&rft.genre=book&rft.isbn=9783662575017&rft.pages=101&rft.place=Berlin+%2F+Heidelberg&rft.pub=Springer+Spektrum" style="display:none">

</li>

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