Is $x^\pi$ continuous on $\mathbb{R}$? For real-valued functions such as $x^2$ or $x^\sqrt{2}$ we can show that they are continuous using $\epsilon-\delta$ argument, because the exponent is an algebraic number.
But what if say $f(x):=x^\pi$, or $f(x):=x^e$ ? how can we rigorously prove that they are continuous on $\mathbb{R}$?

For $x>0$, If we take $f(x)=e^{\pi\log(x)}$ and plug it into the MacLaurin Series for the exponential function, we see that it is uniformally convergent, which implies continuity. Same argument can be applied to $x^e$.
EDIT: This might help too https://en.wikipedia.org/wiki/Uniform_limit_theorem
