If f(x)= l x for x 0 0 for x > 0 . then f(x) at x=0 is - Vidyadip

MCQ

If $f(x)=\left\{\begin{array}{l}x \text { for } x \leq 0 \\ 0 \text { for } x > 0\end{array}\right.$ then $f(x)$ at $x=0$ is

✓
Continuouss but not differentiable
B
Not continuous but differentiable
C
Continuous and differentiable
D
Not continuous and not differentiable

✓

Answer

Correct option: A.

Continuouss but not differentiable

$\because \lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0} x=0$

and $\lim _{x \rightarrow 0^{+}} f(x)=0, f(0)=0$

$\therefore f(x)$ is continuous at $x=0$.

For differentiability, $f^{\prime}(x)=\left\{\begin{array}{ll}1 & x \leq 0 \\ 0 & x>0\end{array}\right.$

$\therefore f(x)$ is not differentiable.

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