Constant Function is 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 continuous on $\R$.


Proof

Follows directly from:

Constant Real Function is Uniformly Continuous
Uniformly Continuous Real Function is Continuous.

$\blacksquare$


Sources

  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): continuous function (v)
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