Question

Let $f\left(x\right)=\{\begin{array}{c}(x-1)|x-1|,x\ne 1\\ 0,x=1\end{array}$. Discuss the continuity and differentiability of $f\left(x\right)$ at $x=1$

• A. not Continuous and not differentiable at $x=1$
• B. Continuous and not differentiable at $x=1$
• C. Continuous and differentiable at $x=1$
• D. none of these

Answer 1

Answer: (C)

Continuous and differentiable at $x=1$

$f(1-h)=\underset{h\to 0}{lim}-h\left(h\right)=0$
$f(1+h)=\underset{h\to 0}{lim}h\left(h\right)=0$
$f^{\prime }(1-h)=\underset{h\to 0}{lim}\frac{-h\left(h\right)}{h}=0$
$f^{\prime }(1+h)=\underset{h\to 0}{lim}\frac{h\left(h\right)}{-h}=0$
Hence, function is continuous as well as differentiable at $x=1.$