On the smoothness condition in Euler's theorem on homogeneous functions

Abstract

ABSTRACTFor a function with continuous first partial derivatives, a theorem of Euler characterizes when f is a homogeneous function. This note determines whether the conclusion of Euler's theorem holds if the smoothness of f is not assumed. An example is given to show that if n ≥ 2, a homogeneous function (of any degree) need not be differentiable (and so the conclusion of Euler's theorem would fail for such a function). By way of contrast, it is shown that if n = 1, a homogeneous function (of any degree) must be differentiable (and so Euler's theorem does not need to assume the smoothness of f if n = 1). Additional characterizations of homogeneous functions, remarks and examples illustrate the theory, emphasizing differences in behaviour between the contexts n ≥ 2 and n = 1. This note could be used as enrichment material in calculus courses and possibly some science courses.