What Uniform Convergence Preserves

Real Analysis · Axiom Academy

LESSON What Uniform Convergence Preserves If uniformly and each f_n is nice, the limit f inherits it — continuity, the integral, and boundedness all survive. Pointwise convergence guarantees none of them. 1. Uniform vs Pointwise: the Worst Gap Both modes ask f_n(x) to approach f(x) . The difference is whether the approximation gets good everywhere at once . Uniform convergence controls the single largest vertical gap between f_n and f across the whole domain — the supremum norm — and forces it to shrink to 0 . The proof is the famous argument . Fix a point and a nearby point y . Split the gap |f(x) - f(y)| into three: from f to f_N at x , the continuity of f_N between x and y , and from f_N back to f at y . Uniform convergence kills the two outer pieces — by the same N everywhere — and continuity of f_N kills the middle. The reason is a one-line estimate. The uniform bound holds at every x , so the total area between the two graphs can be no larger than that bound times the width of the interval: Pick N with . Since f_N is bounded, say , the triangle inequality gives for all x . A single finite ceiling holds across the whole domain. 5. Why Pointwise Fails — and the Derivative Caveat Drop the "uniform" and every guarantee can break. The classic witness is f_n(x) = x^n on [0,1] : each one is continuous, yet the pointwise limit is 0 on [0,1) and 1 at x=1 — a discontinuous step. The worst gap never shrinks; convergence here is pointwise but not uniform.

This is the written version of the interactive lesson above. See the full Real Analysis course.