Let $f: \mathbb{R} \to \mathbb{R}$ be continuous at $0$ and $f(x+y) = f(x) + f(y)$ for all $x, y$ in $\mathbb{R}$. Show that there is a real number $c$ such that $f(x) = cx$.
I'm not sure about how to do show this, I can tell that $f(0) = 0$ and that from continuity at $0$ we get $|f(x)| < \epsilon$ for $x$ in some delta neighborhood of $0$. Can anyone help me with this please.
Edit: please note that the function is only continuous at $0$ so Cauchy functional equation properties can't be used here.
