Homogeneous Functions and Euler's Theorem
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From: Maury Barbato on 13 Jul 2010 01:18 Hello, let A be an open cone in R^n (that is, A is an open subset of R^n and tx is in A for every x in A and every real t > 0). I have two problems in my minds, related to the well-known Euler's Theorem about homogeneous function. (I) Let f:A -> R be a homogeneous function of degree p (that is f(tx) = (t^p)*f(x) for every x in A and t > 0) and suppose that the partial derivatives D_j f, 1 <= j <= n, exist for every x in A. Is f differentiable in A? (II) Let f:A -> R be a function such that the partial derivatives D_j f, 1 <= j <= n, exist for every x in A, and suppose that sum_{j = 1 to n} [(D_j f)(x)]*x_j = p*f(x), for every x=(x_1,...,x_n) in A. Is f homogeneous with degree p in A? I think the answer is not to both the questions, but I have no actual counterexample for now. Thank you very much for your attention. My Best Regards, Maury Barbato PS Note that in Euler's Theorem one suppose that f is a differetiable function, so the answer to (II) is not trivial. |