Question

Give an example of a function which is continuos but not differentiable at at a point.

Answer 1

Consider a function, $f\left(x\right)=\left\{\begin{array}{l}x,x>0\\ -x,x\le 0\end{array}\right.$
This mod function is continuous at x=0 but not differentiable at x=0.

Continuity at x=0, we have:

(LHL at x = 0)

$$\begin{gathered} \underset{x\to 0^{-}}{\lim }f\left(x\right) \\ =\underset{h\to 0}{\lim }f(0-h) \\ =\underset{h\to 0}{\lim }-(0-h) \\ =0 \end{gathered}$$

(RHL at x = 0)

$$\begin{gathered} \underset{x\to 0^{+}}{\lim }f\left(x\right) \\ =\underset{h\to 0}{\lim }f(0+h) \\ =\underset{h\to 0}{\lim }(0+h) \\ =0 \end{gathered}$$

and $f\left(0\right)=0$

Thus, $\underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{x\to 0^{+}}{\lim }f\left(x\right)=f\left(0\right).$

Hence, $f\left(x\right)$ is continuous at $x=0.$

Now, we will check the differentiability at x=0, we have:

(LHD at x = 0)

$$\begin{gathered} \underset{x\to 0^{-}}{\lim }\frac{f\left(x\right)-f\left(0\right)}{x-0} \\ =\underset{h\to 0}{\lim }\frac{f(0-h)-f\left(0\right)}{0-h-0} \\ =\underset{h\to 0}{\lim }\frac{-(0-h)-0}{-h} \\ =-1 \end{gathered}$$

(RHD at x = 0)

$$\begin{gathered} \underset{x\to 0^{+}}{\lim }\frac{f\left(x\right)-f\left(0\right)}{x-0} \\ =\underset{h\to 0}{\lim }\frac{f(0+h)-f\left(0\right)}{0+h-0} \\ =\underset{h\to 0}{\lim }\frac{0+h-0}{h} \\ =1 \end{gathered}$$

Thus, $\underset{x\to 0^{-}}{\lim }f\left(x\right)$ ≠ $\underset{x\to 0^{+}}{\lim }f\left(x\right)$

Hence $f\left(x\right)$ is not differentiable at $x=0$.