If $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}\mathrm{x}, & \text { for } \mathrm{x} \leq 0 \\ 0, &…
If $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}\mathrm{x}, & \text { for } \mathrm{x} \leq 0 \\ 0, & \text { for } \mathrm{x}>0\end{array}\right.$, then the function $\mathrm{f}(\mathrm{x})$ at $\mathrm{x}=0$ is
not continuous and not differentiable
not continuous but differentiable
continuous but not differentiable
continuous and differentiable
Solution
$\begin{aligned}
& \mathrm{f}(\mathrm{x})=\mathrm{x}, \quad \text { if } \mathrm{x} \leq 0 \\
& =0, \quad \text { if } x>0 \\
& \therefore \quad \lim _{x \rightarrow 0^{-}} f(x)=\lim _{n \rightarrow 0} x=0 \text { and } \lim _{x \rightarrow 0^{+}} f(x)=0 \\
& \mathrm{f}(0)=0 \\
& \mathrm{f}^{\prime}(\mathrm{x})=1, \quad \text { if } \mathrm{x} \leq 0 \\
& =0, \quad \text { if } x>0 \\
&
\end{aligned}$
Thus $f(x)$ is continuous at $x=0$
$\begin{aligned}
\mathrm{f}^{\prime}(\mathrm{x}) & =1, & & \text { if } \mathrm{x} \leq 0 \\
& =0, & & \text { if } \mathrm{x}>0
\end{aligned}$
Thus $\mathrm{f}(\mathrm{x})$ is not differentiable at $\mathrm{x}=0$.