Thread: Invertible function problem

Q: Suppose that the function g: R->R is continuous and that g(x) > 0 for all x in R. Define h:R->R by int (upper limit = x, lower = 0) 1/g(t). Note that h is strictly increasing on R, hence h is invertible. Set J = rang h and let f:J->R be inverse of h.

Prove that f satisfies the differential equation:

f'(x) = g(f(x), for all x in J
f(0) = 0