1. The problem statement, all variables and given/known data Suppose f ' is continuous on [a, b] and ε > 0. Prove that there exists ∂ > 0 such that | [f(t)-f(x)]/[t-x] - f '(x) | < ε whenever 0 < |t - x| < ∂, a ≤ x ≤ b, a ≤ t ≤ b. 2. Relevant equations Definitions of continuity and differentiability 3. The attempt at a solution Fix x in [a, b] and ε > 0. Since f ' is continuous at x, there exists a ∂ > 0 such that |f '(x) - f '(t) | < ε whenever 0 < |x - t| < ∂, a ≤ t ≤ b. Now I'm not sure how this gets me | [f(t)-f(x)]/[t-x] - f '(x) | < ε. I've tried a lot of different things. Can you give me an itsy bitsy hint?