Differentiability Class/Examples/Class 0 Function with Derivative Discontinuous at Point

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

Proof
For $x = 0$:

For $x \ne 0$:

From Differentiable Function is Continuous, $f \in C^0$.

$f'$ is not continuous at $0$, so $f \notin C^1$.