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1994•Proceedings of the American Mathematical SocietyOpen access

Locally Uniformly Continuous Functions

Alexander J. Izzo

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Abstract

It is shown that on every infinite-dimensional separable normed space there exist continuous real-valued functions that are nowhere locally uniformly continuous. An explicit example of such a function on ${l^p}\;(1 \leq p < \infty )$ is given. It is also shown that every continuous real-valued function on a metric space can be approximated uniformly by locally uniformly continuous functions.

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It is shown that on every infinite-dimensional separable normed space there exist continuous real-valued functions that are nowhere locally uniformly continuous. An explicit example of such a function on ${l^p}\;(1 \leq p < \infty )$ is given. It is also shown that every continuous real-valued function on a metric space can be approximated uniformly by locally uniformly continuous functions.

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Available abstract

It is shown that on every infinite-dimensional separable normed space there exist continuous real-valued functions that are nowhere locally uniformly continuous. An explicit example of such a function on ${l^p}\;(1 \leq p < \infty )$ is given. It is also shown that every continuous real-valued function on a metric space can be approximated uniformly by locally uniformly continuous functions.

Key concepts: Uniform continuity, Separable space, Mathematics, Uniform limit theorem, Continuous function (set theory), Metric space, Continuous functions on a compact Hausdorff space, Space (punctuation)

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