Uniform continuity
Objective. Distinguish pointwise and uniform control.
- 1. Learn
- 2. Worked example
- 3. Practice
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Learn: the key idea
Uniform continuity strengthens ordinary continuity by requiring one δ that works for all points simultaneously: for every ε > 0 there is δ > 0 such that |x − y| < δ implies |f(x) − f(y)| < ε for all x, y in the domain. Pointwise continuity only demands this near each fixed point, with δ possibly shrinking as the point varies. The function f(x) = 1/x on (0, 1) is continuous but not uniformly continuous, since δ must shrink without bound as x → 0. By contrast, any function continuous on a closed bounded interval [a, b] is automatically uniformly continuous, by the Heine-Cantor theorem.
Worked example
Explain why f(x) = 1/x fails uniform continuity on the interval (0, 1), using specific points.
- 1Take xₙ = 1/n and yₙ = 1/(n+1), so |xₙ − yₙ| = 1/(n(n+1)) → 0.
- 2Compute |f(xₙ) − f(yₙ)| = |n − (n+1)| = 1.
- 3So points can be arbitrarily close while outputs differ by a fixed amount 1.
- 4No single δ can force |f(x) − f(y)| < 1 for all such close pairs, so uniform continuity fails.
Answer: f(x) = 1/x is not uniformly continuous on (0, 1)
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Start practisingCommon mistake. Students often confuse continuity at each point with uniform continuity on the whole domain; f(x) = x² is continuous everywhere but not uniformly continuous on ℝ, since the required δ shrinks as x grows large.
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