Constant Function is Uniformly Continuous/Real Function

Theorem

Let $f_c: \R \to \R$ be the constant mapping:

$\exists c \in \R: \forall a \in \R: \map {f_c} a = c$

Then $f_c$ is uniformly continuous on $\R$.


Proof

Follows directly from:

Constant Function is Uniformly Continuous: Metric Space
Real Number Line is Metric Space.

$\blacksquare$


Sources

  • 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $8.2$: Definition and examples: Remarks $8.2.4 \ \text{(a)}$
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