calculus - Let $f:mathbb{R} to mathbb{R}$ be continuous satisfying: $f(x+y) = f(x) + f(y)$. Show $f$ is linear

Let $f:\mathbb{R} \to \mathbb{R}$ be continuous satisfying: $f(x+y) = f(x) + f(y)$. Show $f$ is linear.

What I thought would be simple and still probably is, I'm having trouble with. So to show this function is linear, I have to show it satisfies two properties:

1) for any $x+y \in \mathbb{R}$ that $h(x+y) = h(x) + h(y)$

This one is satisfied, just by definition of the function.

2) for any $x \in \mathbb{R}$ and scalar $c \in \mathbb{R}$ then $h(cx) = ch(x)$

I'm having trouble proving this. Since there is no explicit function which would make this easier I'm left struggling how to apply continuity. I thought perhaps if I said:

"suppose $f$ is continuous at the point $a$. Then by definition:

$\forall \epsilon > 0$ and $\forall a >0$ there exists $\delta > 0$ such that if $|x-a| < \delta \rightarrow |f(x) - f(a)|< \epsilon$.

So I thought if I adapt this definition and instead say $\forall \epsilon > 0$ and $\forall a >0$ there exists $\delta > 0$ such that if $|cx-ca| = |c||x-a| < \delta \rightarrow |c||f(x) - f(a)|< \epsilon$.

I'm not sure that really says anything at all, but it is what i came up with.

Advice on how to proceed?

Answer

First try to prove that $f(nx) = nf(x)$ for integers $n$, then do the same for rational numbers by similar methods, then you can conclude the result by continuity.