Find all continuous function $f: \mathbb R \rightarrow \mathbb R$

Hint: With $g(x):=f(x)+2$, we have$$ g(x+y)=g(x)+g(y).$$

As transformed by @Hagen von Eitzen into the Cauchy functional equation (https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation). The only continous solution to Cauchy's functional equation is $g(x)=kx$. Therefore, $f(x)+2=kx$. Now given that $f(3)=5$ so that $k=\frac{7}{3}$. Thus the required function is $f(x)=\frac{7}{3}x-2$.