Question

Sketch the graph of a function defined everywhere satisfying f'(x) = 0 for every x, except at x = 0 where the derivative does not exist.

Answer 1

derivatives do not exist at cusp and corners. Are you missing any other information?
f '(x)=0 means the derivative of f is zero

Answer 2

No, that's it. Any graph that will satisfy this will do. I assume that a graph that is horizontal the whole way (thereby having no slope) would satisfy this...I think I could make it y=1 for x's less than zero and y=-1 for x's greater than zero to satisfy the question. Sound plausible?

Answer 3

yes because in order for the limit to exist the right hand and left hand limits must be the same. When you take the derivative of each function, it is zero.
Yes, I think this will work