Power Function on Strictly Positive Base is Continuous/Real Power

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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