Differentiation class

The differentiation class is a term from mathematics , especially from the sub-area of analysis . It is a function space and includes all functions that are continuously differentiable at least times , where is a natural number . The differentiation class is usually noted using . ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle C^{k}}$

definition

Let be a number and a non-empty, open subset of the real numbers . A continuous function then belongs to the differentiation class or, more precisely , if it is continuously differentiable to at least- times . ${\displaystyle k\in \mathbb {N} \cup \{0\}}$ ${\displaystyle D\subset \mathbb {R} }$ ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle C^{k}(D)}$ ${\displaystyle C^{k}(D,\mathbb {R} )}$ ${\displaystyle f}$ ${\displaystyle D}$ ${\displaystyle k}$

According to the definition, the class of continuous functions and the differentiation class of functions that can be differentiated as often as desired are referred to. ${\displaystyle C(D):=C^{0}(D)}$ ${\displaystyle C^{\infty }(D)}$

Generalizations

The class of the analytic functions is sometimes referred to by analogy to the definition above . ${\displaystyle C^{\omega }(D)}$

The definition is adopted analogously for continuous functions in the multidimensional Euclidean vector space . The function therefore belongs to the differentiation class if it is continuously differentiable over at least a number of times . ${\displaystyle g\colon {\tilde {D}}\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ ${\displaystyle g}$ ${\displaystyle C^{k}({\tilde {D}},\mathbb {R} ^{m})}$ ${\displaystyle {\tilde {D}}}$ ${\displaystyle k}$

If the number of possible differentiations ( ) for multi-dimensional functions differs between the individual variables, this can be taken into account in a generalization of the above notation: ${\displaystyle k,l,..}$ ${\displaystyle C^{k,l,..}(D).}$

The differentiation classes are also defined analogously for functions between differentiable manifolds . ${\displaystyle C^{k}}$

Subset relation

Let be an open subset, then ${\displaystyle D\subset \mathbb {R} ^{n}}$

${\displaystyle C^{\omega }(D)\subset C^{\infty }(D)\subset \dotsb \subset C^{k}(D)\subset \dotsb \subset C^{1}(D)\subset C^{0}(D)}$.

The higher the index of the differentiation class, the fewer functions it comprises. ${\displaystyle k}$

Examples

• The exponential function is analytical and therefore belongs to the class . ${\displaystyle \exp \colon \mathbb {R} \rightarrow \mathbb {R} }$ ${\displaystyle C^{\omega }(\mathbb {R} )}$
• The amount function is continuous, but not differentiable. So it belongs to the class , but not to the class . ${\displaystyle |\cdot |\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle C^{0}(\mathbb {R} )}$ ${\displaystyle C^{1}(\mathbb {R} )}$
• The function , is twice continuously differentiable, but not three times. So it applies . ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle x\mapsto |x|^{3}}$ ${\displaystyle f\in C^{2}(\mathbb {R} )\setminus C^{3}(\mathbb {R} )}$
• The function with for and can be differentiated any number of times and thus belongs to the class , but it is not analytical. ${\displaystyle g\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle g(x)=\exp \left(-{\tfrac {1}{x^{2}}}\right)}$ ${\displaystyle x\neq 0}$ ${\displaystyle g(0)=0}$ ${\displaystyle C^{\infty }(\mathbb {R} )}$
• The function with for and is differentiable everywhere, but the derivative function is not continuous at zero. Thus the function does not belong to the class , only to the class . ${\displaystyle h\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle h(x)=x^{2}\sin(1/x)}$ ${\displaystyle x\neq 0}$ ${\displaystyle h(0)=0}$ ${\displaystyle C^{1}(\mathbb {R} )}$ ${\displaystyle C^{0}(\mathbb {R} )}$

Sufficiently smooth

In connection with differentiability, it is sometimes said that a function is sufficiently smooth . This means that it is often differentiable in the respective context, so you don't have to worry about the differentiability, so to speak. The term is derived from the term smooth function for a function that can be differentiated as often as required.

Individual evidence

1. ^ Rolf Walter: Introduction to Analysis. Volume 3. Walter de Gruyter, Berlin a. a. 2009, ISBN 978-3-11-020960-0 , pp. 59, 147ff.
2. ^ ^{A b} Konrad Königsberger : Analysis 1. Springer-Verlag, Berlin a. a., 2004, ISBN 3-540-41282-4 , p. 155.
3. ^ Konrad Königsberger : Analysis 2. Springer-Verlag, Berlin / Heidelberg, 2000, ISBN 3-540-43580-8 , p. 62.
4. ^ Rolf Walter: Introduction to Analysis 2 . de Gruyter, 2007, ISBN 978-3-11-019540-8 , p. 64, 448 .
5. ^ Prof. Martin Keller-Ressel: Stochastic Analysis. In: TU Dresden, Faculty of Mathematics. May 23, 2015, p. 46 , accessed January 9, 2020 . 
6. ↑ Dirk Langemann, Cordula Reisch: Mathematics is that simple - Partial differential equations for users . 1st edition. Springer Spectrum, Berlin / Heidelberg 2018, ISBN 978-3-662-57501-7 , p. 101 .