Question

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

Answer 1

F is continuous every-where. But it is not differentiable at two points. Note it is not differentiable at x = 0 and x = 1! I mean just consider f(x)= ∣x∣+∣x − 1∣
If you need me to expand let me explain it a bit more.
If you define a function f(x) so that f(x) = |x| for x<0. f(x) = sinx for 0 ≤ x < 22/7, We have f(x) = |x - 22/7| for x ≥ 22/7 then we can say this function is continuous however it will still have two points that are not differentiable. This would be for |x|