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.

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