Question

Does there exist a function which is continuous every where but not differentiable at exactly two points? Justify your answer.

Answer 1

Yes, there exist such a function

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

As we can see in the graph of this function, this function is continuous at every point.

But not differentiable at (1,1) and (2,1) because of a sharp turn.