Definition:Differentiability Class

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Definition

Let $f: \R \to \R$ be a real function.

Then $\map f x$ is of differentiability class $C^k$ if and only if:

$\dfrac {\d^k} {\d x^k} \map f x \in C$

where $C$ denotes the class of continuous real functions.


That is, $f$ is in differentiability class $k$ if and only if there exists a $k$th derivative of $f$ which is continuous.


If $\dfrac {\d^k} {\d x^k} \map f x$ is continuous for all $k \in \N$, then $\map f x$ is of differentiability class $C^\infty$.


Specific Instances

Class Zero

Differentiability class $C^0$ consists of the class of continuous real functions $C$ whether they be differentiable or not.


Continuously Differentiable

A differentiable function $f$ is continuously differentiable if and only if $f$ is of differentiability class $C^1$.

That is, if the first order derivative of $f$ (and possibly higher) is continuous.


Smooth Function

A real function is smooth if it is of differentiability class $C^\infty$.

That is, if it admits of continuous derivatives of all orders.


Domain Restriction

Let $f: \R \to \R$ be a real function.

Let $S \subseteq \R$ be a subset of $\R$ on which the $n$th derivative of $f$ is continuous on $S$.


Then $\map f x$ is of differentiability class $C^n$ on $S$.


Also known as

A real function in differentiability class $C^n$ can be described as being $n$ times differentiable.

Some authors use $C^{\paren n}$ for $C^n$.


Examples

Class $C^0$ Function

Let $f$ be the real function defined as:

$\map f x = \begin {cases} 0 & : x < 0 \\ x & : x \ge 0 \end {cases}$

Then $f \in C^0$ but $f \notin C^1$.


Class $C^1$ Function

Let $f$ be the real function defined as:

$\map f x = \begin {cases} 0 & : x < 0 \\ x^2 & : x \ge 0 \end {cases}$

Then $f \in C^1$ but $f \notin C^2$.


Class $C^n$ Function

Let a real function $f$ be required that has the following properties:

$(1): \quad f \in C^n$
$(2): \quad f \notin C^{n + 1}$

where $C^k$ denotes the differentiability class of order $k$.


Then $f$ may be defined as:

$\map f x = \begin {cases} 0 & : x < 0 \\ x^{n + 1} & : x \ge 0 \end {cases}$


Class $C^0$ Function with No Derivative at Point

Let $f$ be the real function defined as:

$\map f x = \begin {cases} x^2 \sin \dfrac 1 x & : x \ne 0 \\ 0 & : x = 0 \end {cases}$

Then $f \in C^0$ but $f \notin C^1$.

The latter is because $f'$ does not exist at $x = 0$.


Also see

  • Results about differentiability classes can be found here.


Sources