# Euler's Homogeneous Function Theorem

I have to proof Euler's Homogeneous Function Theorem, i.e. for a function $$f:\mathbb{R}^n\setminus\{0\}\rightarrow \mathbb{R}$$ and $$\alpha\in\mathbb{R}$$ I have to show the the equivalence of:

1. $$f(tx)=t^{\alpha}f(x)\quad(\forall x\neq 0,\forall t\in\mathbb{R}_+)$$
2. $$\nabla f(x)\cdot x=\alpha f(x)$$

This follows directly from the multidimensional chain rule. But the problem is I am only allowed to use the one-dimensional chain rule. I had success with showing the implication $$1.\rightarrow 2.$$ but I can't make the other one. I think I need to work in a proof of a special case of the multi-dimensional chain-rule but I always get stuck. Can somebody help or does someone know another way?

• You are equating a scalar and a vector. I think you mean $\nabla f(x)\cdot x=\alpha f(x)$ – Rafa Budría Jul 12 at 11:01
• edited, thank you! – Jules Jul 12 at 12:33