Real function/Exponential equation/Continuous/Exponential function/Exercise

Let

$f\colon \mathbb {R} \longrightarrow \mathbb {R}$

be a continuous function ${}\neq 0$, with the property that

${}f(x+y)=f(x)\cdot f(y)\,$

for all ${}x,y\in \mathbb {R}$. Prove that ${}f$ is an exponential function, i.e. there exists a ${}b>0$ such that

${}f(x)=b^{x}$.

Create a solution