(Edited) Let $f:R\to R^+$ a continuous function where $f(x)f(-x)=1$ for every $x \in R$. Is $f$ necessarily the exponential?

Question

Additionally, we suppose $f'(x)\neq 0$ in any point, so it is not constant at any interval.

Answer 1

Take any function $f$ on $[0,\infty)$ with $f(0)=1$ that doesn't take the value $0$, and define $f$ on $(-\infty, 0)$ by $f(x) = 1/f(-x)$.

Answer 2

We have this result if $f$ is continuous: Prove if $f(x+y)=f(x)f(y)$ then $f(x)=a^x$. For $f$ not necessarily continuous, consider for example:

$$f(x) = \begin{cases}1 & x \in \mathbb Q \\ -1 & x \not \in \mathbb Q\end{cases}$$

since $x$ is rational iff $-x$ is.