Suppose $f: \mathbb{R} \to \mathbb{R}$ is differentiable. I think it need Euler's theorem to solve it. But I do not know the homogeneous degree of g. My attempt is $$g(tx,ty)=f(\sqrt {(tx)^2+(ty)^2})=f(|x|\sqrt {x^2+y^2})$$ Another question is can I say g (two variable function) is differentiable since f (one variable function) is differentiable?
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1All you have to do is calculate the partial derivatives. No theorem on homogeneous functions is needed. – Kavi Rama Murthy May 18 '20 at 06:15
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@KaviRamaMurthy Thanks! – Steven Lu May 18 '20 at 08:11
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Concerning your other question, recall that a partial derivative is just an ordinary derivative by one variable while all the other independent variables are held constant. And by the ordinary chain rule, the composition of two differentiable functions is differentiable. – Paul Sinclair May 18 '20 at 13:55