Power Function on Strictly Positive Base is Continuous

Theorem

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

Rational Power

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

$\map f x = a^x$

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

Then $f$ is continuous.

Real Power

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.