Differentiability Class/Examples/Class Zero Function

Example of Differentiability Class
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 $f \notin C^1$.

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

We have that:
 * $\map {f'} x = \begin {cases} 0 & : x < 0 \\ 1 & : x \ge 0 \end {cases}$

Hence $f'$ is not continuous at $x = 0$.

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