f(x) = √x Uniformly Continuous on [0,∞)

Real Analysis · Axiom Academy

EXAMPLE Uniform Continuity of on Proving one works everywhere on an unbounded domain, straight from a square-root inequality. Prove that is uniformly continuous on . That is: for every there is a single — depending on alone, not on the points — such that for all , if then . Why one can cover the whole half-line As x grows, the graph of gets flatter, so a fixed horizontal window of width never lifts the output by more than — even though the domain runs to infinity. The key inequality below makes that exact. Nicely done — you proved is uniformly continuous on the whole half-line with a single . The essentials: One inequality did the work: holds for all — bounding the output gap by something that depends only on the input gap. depends on alone: taking makes , and that choice never mentions x or y — which is exactly what "uniform" demands. Unbounded domain, still uniform: the slope blows up near 0 , yet the function flattens as ; the square-root inequality tames both ends at once, so no point needs its own . Contrast g(x)=x^2 on : it is not uniformly continuous, because for a fixed gap |x-y| the output gap grows without bound as the points move out. Square root flattens; squaring steepens.

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