`` A Theorem of Joris''
                               J. Duncan  

 ABSTRACT. In its simplest (and most interesting) form, Joris' Theorem says that if f is a real function on the real line with f^2 and f^3 infinitely differentiable then f is also infinitely differentiable.  The original proof is rather formidable.   An easy proof was recently published in the Monthly, but it turned out to be bogus.   We shall present the proof by Amemiya & Masuda which is in the spirit of commutative algebra.