f(x) = x² Not Uniformly Continuous on ℝ

Real Analysis · Axiom Academy

EXAMPLE f(x) = x^2 Is Not Uniformly Continuous on A continuous function can still fail uniform continuity on an unbounded domain — we disprove it with two sequences. The function f(x) = x^2 is continuous at every real number. Prove that it is nonetheless not uniformly continuous on all of . (Recall: f is uniformly continuous on S if for every there is a single such that for all .) Two inputs the same small distance apart land close together near the origin, but far apart out where the parabola is steep. The slope 2x has no ceiling, so no single controls the output everywhere. Nice work. You disproved uniform continuity by exhibiting two sequences whose inputs collapse together while their outputs stay 2 apart. Continuous ≠ uniformly continuous: f(x) = x^2 is continuous at every point, yet not uniformly continuous on the unbounded domain . The witness: with x_n = n and the inputs satisfy , but , which does not . Why has no : for any candidate , choosing n large makes while the output gap stays — the definition is violated. Root cause: the slope f'(x) = 2x is unbounded, so the needed near a point shrinks as x grows; on a bounded interval [a,b] the slope is capped and f is uniformly continuous. This sequence-pair technique — find inputs that converge while their images stay apart — is the standard way to disprove uniform continuity, and it generalizes to any function whose rate of change blows up on an unbounded set.

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