Question

If $f\left(x\right)=\{\begin{array}{cc}x,& \text{for}x\le 0\\ 0,& \text{for}x>0\end{array}$

then $f\left(x\right)$ 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

The correct option is A Continuous but not differentiable

Continuity at $x=0$
$\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0}{lim}x=0$
$\underset{x\to 0^{+}}{lim}f\left(x\right)=0$
$f\left(0\right)=0$
$\therefore$ continuous at $x=0$
For differentiablility
$f^{\prime }\left(x\right)=\{\begin{array}{c}1,x\le 0\\ 0,x>0\end{array}$
$\therefore$ not differentiable
It has sharp edge at $x=0$
$\therefore$ not differentiable but continuous.