analysis - Continuity of a function

For all $(x,y)\in\mathbb{R}^2$ a function $f$ satisfies $f (x+y)=f (x)\cdot f (y)$.

If $f$ is continuous at a point $a$, then show that $f$ is continuous on $\mathbb R$ and $f (x)= b^x$ for some constant $b$.

My approach

$$f (1)=f (1+0)=f (1)\cdot f (0)$$
Thus $f(0)=1$ and $$f(x+h)-f (x)= f(x)\,(f (h)-1)$$

I can prove the continuity of the function.
How can I prove that $f (x)=b^x$ for any $x\in\Bbb R$?