Let $f: R^n \to R$ be a continuous function on $R^n$ that is homogeneous of degree 1. Suppose that $f$ is differentiable at the origin $(0, 0)$. Prove that $f$ is a linear transformation.
This is what I have so far. In order to show f is a linear transformation it must satisfy two conditions.
1) $f(t\cdot x) = t\cdot f(x)$, this is satisfied by f as it is homogenous of degree 1.
2) $f(x+y) = f(x) + f(y)$, I have been told to prove this by using the fact that
$$(Df)_pu = \lim (1/t)\cdot(f(p + tu) − f(p)) \quad\text{as}\quad t\to0$$
Can anyone show me how to arrive at this conclusion from this?
HINT. Differentiate termwise the equation $$ f(tx)=tf(x), $$ where $x\in\mathbb{R}^n$ is fixed.