if $f(x + y) = f(x)f(y)$ is continuous, then it has to be injective.
Solution 1:
Hint : show that $f(t)=f(\frac{t}{2})^2$. Deduce $f\geq 0$. Let $G=\lbrace x\in{\mathbb R} | f(x)=1\rbrace$. Then $G$ is an additive subgroup of $\mathbb R$. If $f$ is continuous it is also a closed subgroup. Also, show that $t\in G \Rightarrow \frac{t}{2} \in G$ using the above identity. Deduce that $G$ is dense if $G\neq \lbrace 0 \rbrace$. Conclude.