Suppose $f(k) = x^k$ is a function mapping $mathbb{Q}tomathbb{R}$ where $k$ is rational and $x>0$. If we can prove continuity at a single point in $mathbb{Q}$ is it necessarily the case $f$ will be continuous over all $mathbb{Q}$?

Suppose that single point is taken to be $x’$ with $f : mathbb{Q} to mathbb R$. Then,

$$

forall varepsilon > 0, exists delta > 0 quad text{s.t. with} x’ in D, quad left|x-x’right| < delta quad Rightarrow quad |f(x)-f(x’)| < varepsilon

$$

From here, is it possible to extrapolate the structure and ascertain if it will hold over all $mathbb{Q}$?