Differentiability Class/Examples/Class n Function

Example of Differentiability Class
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:


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

Proof
By inspection it is seen that $f$ is continuous everywhere.

We have from Nth Derivative of Mth Power that:
 * $\map {f^{\paren n} } x = \begin {cases} 0 & : x < 0 \\ \paren {n + 1}^{\underline n} x & : x \ge 0 \end {cases}$

where $\paren {n + 1}^{\underline n}$ denotes the $n$th falling factorial of $n + 1$.

At $x = 0$ it is seen that $\map {f^{\paren n} } x = 0$ and so $\map {f^{\paren n} } x$ is continuous at $0$.

By inspection it is seen that $f^{\paren n}$ is continuous everywhere.

So by definition of differentiability class, $f$ is a member of $C^n$.

Then we have that:


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

Hence $f^{\paren {n + 1} }$ is not continuous at $x = 0$.

So by definition of differentiability class, $f$ is not a member of $C^{n + 1}$.