For a function "f": [real numbers set][superscript n]\{(0,…,0)}[right arrow][real numbers set] 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.