real analysis - For given $a in mathbb{R}$ there exists unique continuous function $f:mathbb{R} to mathbb{R}$.

For given $a \in \mathbb{R}$ there exists unique continuous function $f:\mathbb{R} \to \mathbb{R}$ that satiafy $f(x+y)=f(x)f(y)$ for $x,y \in \mathbb{R}$ and $f(1)=a$.

These theorems were discussed in my mathematical-modeling class and were given to us to prove them.