Question

If $f(x)=\begin{cases} x, & \text { for } x \leq 0 \\ 0, & \text { for } x>0 \end{cases}$
then $f(x)$ at $x=0$ is

• A. Continuous but not differentiable
• B. Not continuous but differentiable
• C. Continuous and differentiable
• D. Not continuous and not differentiable

Answer 1

Answer: (A)

Continuity at $x =0$
$\displaystyle\lim _{x \rightarrow 0^{-}} f(x)=\displaystyle\lim _{x \rightarrow 0} x=0$
$\displaystyle\lim _{x \rightarrow 0^{+}} f(x)=0$
$f (0)=0$
$\therefore $ continuous at $x =0$
For differentiablility
$f'(x)=\begin{cases}1, x \leq 0 \\ 0, x>0\end{cases}$
$\therefore $ not differentiable
It has sharp edge at $x=0$
$\therefore $ not differentiable but continuous.