Measure, Integration & Real Analysis · Problem 2E.4

Integration — Sheldon Axler

Mathematics — Measure Theory · uniform-convergence-uniform-continuity

Problem

Prove or give a counterexample: If

A \subseteq \mathbf{R}

and

f_1, f_2, \ldots

is a sequence of uniformly continuous functions from

A

\mathbf{R}

that converges uniformly to a function

f: A \to \mathbf{R}

, then

f

is uniformly continuous on

A