If $e^{f(x)}$ is continuous, does it follow that $f$ is continuous?
I say yes.
Proof by contradiction Suppose not. That is, suppose $e^{f(x)}$ is continuous and $f$ is not continuous. Then $$\lim_{x\to x_0}f(x)\ne f(x_0).$$ Thus, $$\lim_{x\to x_0}e^{f(x)}\ne e^{f(x_0)}.$$ But by definition of continuity, $$\lim_{x\to x_0}e^{f(x)}=e^{f(x_0)}.$$ Therefore, we have a contradiction and if $e^{f(x)}$ is continuous, it follows that $f$ is continuous.
But this proof seems a bit lacking. Am I correct in my assumption? How can I strengthen my proof if I am correct?
You know that $\log$ is a continuous function, right? And $f(x) = \log e^{f(x)}$ for all $x \in {\rm dom}(f)$, so $f$ is a composition of continuous maps, hence continuous.