Uniform Continuity

Intro to Proofs · Axiom Academy

One δ for the whole domain, or a fresh δ at every point? That single swap in the order of the quantifiers separates continuity from uniform continuity. 1. The δ Comes First — or Last? Both definitions juggle the same symbols; only the order of the quantifiers changes. In pointwise continuity you may choose the point before you commit to , so is allowed to depend on it. In uniform continuity you must produce first — it has to work for every pair of nearby points simultaneously. Pointwise: pick c first, so may shrink point by point Uniform: commit to first — one width for the whole domain f(x)=x^2 is continuous at every point, yet not uniformly continuous on . The reason is geometric: the curve steepens without bound, so a horizontal gap that is harmless near the origin becomes a chasm far out. Hold the gap fixed and slide the pair right — the output gap outruns any tolerance you fixed in advance. With that pair the inputs stay closer than , yet the outputs separate by The term grows without bound as , so it eventually exceeds no matter how small the fixed was. A concrete trigger: the gap first passes 1 once — for that is already . No single survives. 3. A Bounded Slope Hands You One δ What rescued x^2 on [0,M] is exactly what makes the difference everywhere: a bounded slope . If on the whole domain, the Mean Value Theorem gives — a single Lipschitz inequality. Now the same fixed tolerance box keeps the same width at every point, and one covers the entire domain.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.