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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?

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

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