Question

Does there exist a function which is continuous everywhere but not differentiable at exactly tow points? Justify your answer.

Answer 1

Yes there exist such a function

$f\left(x\right)=|x-1|+|x-2|$

As you can see the graph of the function, this function is continuous at every point. But it is differentiable at exactly two points, viz $(1,1)$ and $(2,1)$ because of a sharp turn.