Anti Derivative question (it's tricky)...?

Question

Find g(x), f(x), h(x) such that f(x), h(x) are both anti derivatives of g(x). but there is NO constant c such that f(x) = h(x) +c for all x in the domain of g.

Read carefully, and explain.

Answer 1

My initial guess would be:

g(x) = sinxcosx
f(x) = (sinx)^2 / 2 + C
h(x) = -(cosx)^2/2 + C

Answer 2

In the first answer, the two Cs were just 1/2 apart from each other, so that one is obviously wrong.

OK. (f-h)' = g-g = 0, so f-h = constant, assuming that f and h are differentiable everywhere. So the only way you have a chance is to be tricky about the domains.

E.g:

f(x) = 1 if x is negative, -1 if x is positive
h(x) = - f(x)

Both have derivative = 0 for all x not equal to 0, and undefined derivative at 0.