Question

Discuss the continuity & differentiability of function $f\left(x\right)=\left\{\begin{array}{c}2+\sqrt{(1-x^2)},|x|\le 1\\ 2e^{{(1-x)}^2},|x|>1\end{array}\right\}$ ?

Answer 1

$\underset{x\to 1}{lim}f\left(x\right)=2+\sqrt{1-1}=2$

$\underset{x\to 1}{lim}f\left(x\right)=2e^{(1-1{)}^2}=2$

$f\left(1\right)=2$

$\therefore$ It is continues

$\underset{x\to 1}{lim}{f}^{\prime }\left(x\right)=\frac{-2}{2\sqrt{1-x^2}}=\frac{-x}{\sqrt{1-x^2}}$

$\therefore$ Not differentiable at $x=1$