Show $f:[1,\infty[\rightarrow[1,\infty[, x\mapsto x^x$ is bijective and continuous, and show continuity for its inverse function $f^{-1}$.
My idea here is to proceed with (hopefully) minimal effort and argue that $x^x$ is a strictly monotonic function and continuous, which implies both bijectivity and the continuity of $f^{-1}$. However, I'm wondering if this is not too lazy:
We can rewrite $f(x)$ as $\exp^{x\log(x)}$ and $f(x)$ is stricly increasing for every $x$ since all $x\geq1$. Its continuity follows from the fact that $f(x)$ is a composition of continuous functions and these two facts in turn imply the bijectivity of $f$ as well as the continuity of $f^{-1}$ as desired.