Finding the Domain of an undefined function

Question

I tried a lot to solve this problem, ended up in a confusion...

So here's the problem:

Consider $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f\left(x+y\right)=f\left(x\right)f\left(y\right) $.

Then find the Domain of $\frac{1}{\sqrt{f\left(x\right)}}$.

Answer 1

You don't need all that theory to get the result that you want. Just follow the steps below:

$f(x)=0$ is a solution, in which case the domain is the empty set.

If there is an $x_0$ such that $f(x_0)\ne 0$ (for example when $f(x)=a^x$), we have:

$f(x)\ne 0$ for all $x$ since $f(x_0)=f(x)f(x_0-x)$

$f(x) > 0$ for all $x$ since $f(x)=f(x/2)f(x/2)=f(x/2)^2$

So in this case the domain is the whole real line

Answer 2

If f is continuous it's an exponential, so $f(x) = a^x$ for an $a \ge 0$

For $a=0$, so $f\equiv 0$ the letter $\frac{1}{\sqrt{f(x)}}$ is not well defined…