Answer to Question #347211 in Real Analysis for Nikhil

Question

Every continuous function is differentiable.

True or false with full explanation.

Answer 1

False.

Counterexample

The function $f(x)=|x|$ is continuous on $(-\infin, \infin).$

$$\lim\limits_{\Delta x\to0^{-}}\dfrac{f(0+\Delta x)-f(0)}{\Delta x}=\lim\limits_{\Delta x\to0^{-}}\dfrac{-\Delta-0}{\Delta x}=-1$$

$$\lim\limits_{\Delta x\to0^{+}}\dfrac{f(0+\Delta x)-f(0)}{\Delta x}=\lim\limits_{\Delta x\to0^{+}}\dfrac{\Delta-0}{\Delta x}=1$$

$$\lim\limits_{\Delta x\to0^{-}}\dfrac{f(0+\Delta x)-f(0)}{\Delta x}=-1$$

$$\not=1=\lim\limits_{\Delta x\to0^{+}}\dfrac{f(0+\Delta x)-f(0)}{\Delta x}$$

Therefore

$$\lim\limits_{\Delta x\to0}\dfrac{f(0+\Delta x)-f(0)}{\Delta x}$$

does not exist.

Therefore the function $f(x)=|x|$ is not differentiable at $x=0.$