Differentiability Class/Examples

Examples of Differentiability Classes

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 Derivative Discontinuous 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 $f \notin C^1$.