Can the sum of a differentiable and a non differentiable function be differentiable?

Question

Can the sum of a differentiable and a non differentiable function be differentiable?

In one of the solutions to a question, my book's author used the fact that the sum of a differentiable and non differentiable function cannot be differentiable. I am wondering if this is always true or are there some counter examples?

Answer 1

Suppose $f$ is differentiable at point $x_0$ while $g$ isn't. If $f+g$ was differentiable at this point then $g=(f+g)-f$ would be differentiable as well which is a contradiction.