# Constant Function is Continuous/Real Function

< Constant Function is Continuous(Redirected from Constant Real Function is Continuous)

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## 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 Function is Uniformly Continuous: Real Function
- Uniformly Continuous Function is Continuous: Real Function.

$\blacksquare$

## Sources

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