TITLE: Can the sum of a differentiable and a non differentiable function be differentiable?
QUESTION [5 upvotes]: 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?

REPLY [9 votes]: 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.