calculus - Functions $f$ such that $f(x)+f(-x)=f(x)f(-x)$

I was looking for examples of real valued functions $f$ such that $f(x)+f(-x)=f(x)f(-x)$. Preferably, I'd like them to be continuous, differentiable, etc.

Of course, there are the constant functions $f(x)=0$ and $f(x)=2$. I also showed that $1+b^x$, where $b>0$, is another solution. Are there any other nice ones?

Answer

From your solution, I thought to consider the change of variable $f(x) = 1 + g(x)$: your functional equation then becomes

$$ g(x) g(-x) = 1 $$

and now the entire solution space becomes obvious: you can pick $g(x)$ to be any function on the positive reals that is nowhere zero, and then the values at the negative reals are determined by the functional equation. And at zero, you can pick either $g(0) = 1$ or $g(0) = -1$.