Let f, g : [a, b] → R be two continuous functions. If f(x) > g(x) for all x ∈ [a, b], then there exists a number c > 0 such that f(x) ≥ g(x) + c for all x ∈ [a, b].
How should I go about this problem? I have a gut feeling that the IVT can be used in some way. Maybe making a function h = f - g and using that somehow?