Chapter 3, Problem 367RE

True or False? Justify the answer with a proof or a counterexample.

367. Every function has a derivative.

To determine

To show:Justify whether every function has a derivative or not.

Answer to Problem 367RE

False

Explanation of Solution

Formula Used :The definition of derivative is given by, if $f\left(x\right)$ is differentiable at $x=a$, then, $f\text{'}\left(a\right)=\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}$.

Let us consider the counter example, $f(x)=\{\begin{array}{l}x,x\ge 0\\ -x,x<0\end{array}$

Here, f(x) is continuous for all real values of x. But it is not differentiable at x= 0.

  $\begin{array}{l}Rf\text{'}(0)=\underset{h\to 0+}{\lim }\frac{f\left(0+h\right)-f\left(0\right)}{h}\\ =\underset{h\to 0+}{\lim }\frac{h-0}{h}\left[\because f(h)=hash>0\right]\\ =1\\ Lf\text{'}(0)=\underset{h\to 0-}{\lim }\frac{-h-0}{h}\left[\because f(h)=-hash<0\right]\\ =-1\\ \therefore Rf\text{'}(0)\ne Lf\text{'}(0)\end{array}$

Hence, f(x) is not differentiable at x=0.