Continuity Preserved

Real Analysis · Axiom Academy

EXAMPLE Continuity Preserved Under Uniform Limits Use the argument to prove that a uniform limit of continuous functions is itself continuous. Let uniformly on [a,b] , where every f_n is continuous. Prove that the limit function f is continuous at an arbitrary point — and hence continuous on all of [a,b] . Uniform convergence pins f_n inside an strip of f for every x at once; the fixed f_n then carries continuity across the small gap from c to a nearby x . Nicely done — you assembled the argument and proved that uniform convergence preserves continuity. The moves worth keeping: The split: a fixed f_N acts as a bridge between f(x) and f(c) , so |f(x)-f(c)| breaks into three pieces, each forced below . Uniform convergence is the whole point: it hands you ONE N that makes for all x at once — that is why the two outer pieces are small no matter where x sits. Order matters: choose N first (from uniform convergence), then (from continuity of that specific f_N ) — depends on which f_N you fixed. The bound assembles exactly: , so whenever . Pointwise convergence is NOT enough: f_n(x)=x^n on [0,1] has each f_n continuous, yet the pointwise limit is 0 on [0,1) and 1 at x=1 — a jump. The N there must grow as , so the outer pieces can't be controlled uniformly. This is a cornerstone of analysis: uniform limits keep functions "nice." The same bridge reappears throughout the subject whenever you swap one function for a nearby one.

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