4: Riemann Sums, Riemann Integrals, Fundamental Theorem of Calculus

Proposition 1.3. Let X be a subset of R, let x0 be a limit point of X, let f : X → R be a function, and let L be a real number. Then the following two statements are equivalent. • f is differentiable at x0 on X with derivative L. • For every ε > 0, there exists a δ = δ(ε) > 0 such that, if x ∈ X satisfies |x− x0| < δ, then |f(x)− [f(x0) + L(x− x0)]| ≤ ε |x− x0| . Corollary 1.4 (Mean Value Theorem). Let a < b be real numbers, and let f : [a, b]→ R be a continuous function which is differentiable on (a, b). Then there exists x ∈ (a, b) such that f ′(x) = f(b)− f(a) b− a .