Question

following statement is true and which are false? Give reasons for your answers, in the form of a short proof or a counter example.

Every continuous function is differentiable

Answer 1

False.

For example, the function $f(x)=|x|$ is continuous at $x=0$ :

$$\lim\limits_{x\to 0^-}f(x)=0=\lim\limits_{x\to 0^+}f(x)=>\lim\limits_{x\to 0}f(x)=0$$

$$f(0)=|0|=0=\lim\limits_{x\to 0}f(x)$$

On the other hand,

$$\lim\limits_{h\to 0^-}\dfrac{f(0+h)-f(0)}{h}=\lim\limits_{h\to 0^-}\dfrac{|h|-0}{h}=-1$$

$$\lim\limits_{h\to 0^+}\dfrac{f(0+h)-f(0)}{h}=\lim\limits_{h\to 0^+}\dfrac{|h|-0}{h}=1$$

$$\lim\limits_{h\to 0^-}\dfrac{f(0+h)-f(0)}{h}=-1$$

$$\not=1=\lim\limits_{h\to 0^+}\dfrac{f(0+h)-f(0)}{h}$$

Then $\lim\limits_{h\to 0}\dfrac{f(0+h)-f(0)}{h}$ does not exist.

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