True or False: Do Two Anti-Derivatives Necessarily Differ by a Constant? Is it true or false that if $$F'(x) = G'(x)$$ then $$F(x) = G(x) + C.$$ No more details are given (continuity, etc...)
I think this is false but I can't think of an example.
 A: Here's a simpler counterexample, but it may not be the correct solution, depending on subtle details of what is wanted.
Let $f(x) $ be the absolute value function and let $g(x)$ be the function which is $|x|$ for $x\le0$ and $x+1$ for $x>0$.
Then $f$ and $g$ have identical derivatives (both of which are defined if and only if $x≠0$) but do not differ by a constant function.
I would not normally hand over a solution like this, but I judge that it will be just as instructive, in this case, to decide whether the example is a valid answer to the question.
A: This is too long for a comment, and is not a valid counterexample if we want the equality $$F'(x)=G'(x)$$ to hold everywhere and not just almost everywhere (i.e. everywhere except a set of measure zero).
With those caveats in mind, consider the Cantor function:



(The definition is related to the Cantor set, although long enough that I don't care to post it again here.)
Except for a set of measure zero, the derivative of the Cantor function is zero, the same as any constant function. However, the Cantor function is clearly not constant, hence if we allow the derivative to be undefined except on a set of measure zero, then this is a valid counterexample.
