Question: Let I = (a, b) be a nonempty open interval, f is differentiable on I. for each ε > 0, there exists δ > 0 such that for all x, y ∈ (a, b) satisfying 0 < |x − y| < δ we have
$\vert{\frac{f(x)-f(y)}{x-y} - f'(x)}\vert$ < ε
Show that f ′ is uniformly continuous on I.
I think I need to get $\vert{f'(x) - f'(y)}\vert$ < ε from all the given information but I can't figure out how I should proceed.