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Part BCSIR NET June 2023extension-to-closed-interval

Extension to closed interval

Which one of the following functions is uniformly continuous on the interval (0, 1)?

  1. A.f(x) = sin(1/x)
  2. B.f(x) = ✓
  3. C.f(x) = eˣ cos(1/x)
  4. D.

Solution

A continuous function on (0,1) is uniformly continuous iff it extends continuously to [0,1]. → 0 as x → 0⁺, so it extends; the other three oscillate without a limit at 0.

The trap it tests

Pointwise vs uniform

The mode of convergence — or of continuity — was the wrong one.

Drill statements like this

Related counterexample: Continuous on a bounded interval ⇒ bounded

More on this topic

From Continuity and Differentiation › Continuity, uniform continuity, Lipschitz

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