TITLE: Proving that a function of the form f(x+y)=f(x)*f(y) is continuous
QUESTION [5 upvotes]: I have the to solve the following problem:
Let $f$ be a function from the real numbers to the real numbers. The function satisfies $f(x+y) = f(x)f(y)$ for all real $x,y$. Prove that if $f$ is continuous in $0$ then $f$ is continuous in every point.
I think I have a solution but I would like to know if it's correct:
By letting $y=0$ we get that $f(x)=f(x)f(0)$ meaning that $f(0)=1$. Thus we know that for every $\epsilon > 0$ and for every $x$ there exists a $\delta$ such that $|f(x)-1| < \frac{\epsilon}{|f(a)|} $ when $|x| < \delta$. Since this holds for all $x$ we can replace $x$ with $x-a$. Rewriting this we get that whenever $|x-a| < \delta$ we have that $\,|f(a)f(x-a)-f(a)|<\epsilon \implies |f(x)-f(a)| < \epsilon$.

REPLY [3 votes]: Applying lim to both sides as y to 0 $$f(x+y) =f(x)f(y)$$
We get $$\lim_{y\to 0}f(x+y)=\lim_{y\to 0}f(x)f(y)=f(x)\lim_{y\to 0}f(y)=f(x)f(0)=f(x)$$
So by definition $f(x)$ is defined everywhere and the limit exists and is equal $f(x) $. Those 3 facts imply continuity.