An Elementary Proof of Joris's Theorem

Interestingly, Joris's proof equates this property with a different one, and then proceeds by contradiction. Joris's work has been generalized in significant ways in (for instance) [1] and [3]. In [1], Amemiya and Masuda provide a direct proof of Joris's result that is based on ring theory. In this paper, we provide a new proof of Joris's result that utilizes little more than Rolle's theorem, L'Hospital's rule, and Taylor's formula from elementary calculus. Readers who are interested in related problems should also refer to [4], in which Georges Glaeser gives conditions under which a nonnegative function f of class C2m has a square root of class cm; Glaeser's proof is simplified in [2]. In a more recent paper, Joris [6] also takes up a different facet of this question by considering conditions under which a quotient of smooth functions is again smooth. To clarify notation: we write the mth power of a function as f(x)m = H1, f(x) in order to avoid confusion with the mth derivative f(m)(x). Also, f(m)(x)" denotes the nth power of the mth derivative (i.e., [f(m)(x)]n), not the mth derivative of the nth