Definition of Uniform Continuity

Real Analysis · Axiom Academy

LESSON Definition of Uniform Continuity Ordinary continuity lets the tolerance change point by point. Uniform continuity demands one that works across the whole set at once. 1. Ordinary Continuity: Can Depend on the Point Fix a point c and a target tolerance . Continuity at c means we can pick a window of radius around c so that the output never leaves the -band. The catch: at a steeper place, the same window overflows the band, so must shrink . 2. Uniform Continuity: One for the Whole Set Now flip the demand. Choose first — from alone — and require that the same window, of one fixed width, keeps the output inside the -band no matter where it sits on the set. Watch the window slide along f(x)=2x+3 without ever resizing. 3. The Whole Difference Is the Order of Quantifiers Both statements use the same pieces; only the order of and " point" changes. That swap is the entire distinction. On the left the window re-fits at each point; on the right one window is locked and reused. chosen after the point — it may depend on it. chosen before the points — it depends on only. Uniform continuity is therefore the stronger property: every uniformly continuous function is continuous, but not conversely. 4. Example: f(x)=2x+3 Is Uniformly Continuous on The slope is constant, so the output gap is a fixed multiple of the input gap — and that lets one work everywhere. the gap scales by exactly 2 , at every point Given , choose . Then whenever ,

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