Uniformly Continuous Function is Continuous

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Theorem

Metric Space

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_1, d_1}\right)$ be metric spaces.

Let the mapping $f: M_1 \to M_2$ be uniformly continuous on $M_1$.


Then $f$ is continuous on $M_1$.


Real Numbers

Let $I$ be an interval of $\R$.

Let $f: I \to \R$ be a uniformly continuous real function on $I$.


Then $f$ is continuous on $I$.