Question

If $f(x)=\{\begin{array}{ll}x,& \text{ for }x\le 0\\ 0,& \text{ for }x>0\end{array}$
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 $x=0$
$\underset{x\to 0^{-}}{\lim }f(x)=\underset{x\to 0}{\lim }x=0$
$\underset{x\to 0^{+}}{\lim }f(x)=0$
$f(0)=0$
$\therefore$ continuous at $x=0$
For differentiablility
$f^{\prime }(x)=\{\begin{array}{l}1,x\le 0\\ 0,x>0\end{array}$
$\therefore$ not differentiable
It has sharp edge at $x=0$
$\therefore$ not differentiable but continuous.