Suppose f is integrable and F: [a,b] -> R shown by F(x) = ∫_a^x f(t) dt for all x ∈ [a,b]. Prove that F is uniformly continuous. This should help:
Theorem 3.4.4. Let C ⊆ R be a closed bounded interval, and let f: C → R be a function. If f is continuous, then f is uniformly continuous.
Definition 3.4.1. Let A ⊆ R be a set, and let f : A → R be a function. The function f is uniformly continuous if for each ε > 0, there is some δ > 0 such that x, y ∈ A and |x − y| < δ imply |f(x) − f(y)| < ε.
Theorem 5.6.2 (Fundamental Theorem of Calculus Version I). Let I ⊆ R be a non-degenerate interval, let a ∈ I and let f: I → R be a function. Suppose that f|c is integrable for every non-degenerate closed bounded interval C ⊆ I. Let F: I → R be defined by F(x) = ∫_a^x f(t) dt for all x ∈ I. Let c ∈ I. If f is continuous at c, then F is differentiable at c and F'(c) = f(c). If f is continuous, then F is differentiable and F' = f.