Power Function on Strictly Positive Base is Continuous/Real Power

Theorem

Let $a \in \R_{>0}$.

Let $f : \R \to \R$ be the real function defined as:

$\map f x = a^x$

where $a^x$ denotes $a$ to the power of $x$.

Then $f$ is continuous.

Proof

By definition, $a^x$ is the unique continuous extension of $a^r$, for rational $r$.

By definition, continuous extensions are continuous.

Hence the result.

$\blacksquare$

Also see

• Power Function to Rational Power permits Unique Continuous Extension, where such a unique continuous extension is shown to exist.