Calculus Problem on the Exponential Function

A question on one of the math or stat stackexchange sites reminded me of the following freshman advanced calculus problem: if $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function satisfying $f(x)f(y) = f(x+y) \ \forall x,y \in \mathbb{R}$, then $f(x) = e^{\alpha x}$ for some $\alpha \in \mathbb{R}$. I’ll confess that I had to think a little about the problem before coming up with the solution. Since I’ll be teaching advanced calculus to high-school students next year, I thought it best to write up the solution, which can be found here.