PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 39, Number 3, August 1973 AN ISOLATED BOUNDED POINT DERIVATION1 ANTHONY G. O'FARRELL Abstract. For a compact subset X of the plane, R(X) denotes the class of uniform limits on X of rational functions with poles off X. R(X) is a function algebra on X. An example X is constructed such that R(X) admits a bounded point derivation at exactly one point of X. Let X be a compact subset of the plane C. We denote by R(X) the uni- form closure on X of the class R0(X) of rational functions with poles off X. Let x be a point of X. A point derivation on R(X) at x is a linear func- tional D on R(X) such that D(fg)=f(x)Dg + g(x)Df, for every pair/, g of elements of R(X). D is called a bounded point deriva- tion at x if it is continuous when R(X) is given the topology induced by the uniform norm. It is easily seen that a bounded point derivation exists at x precisely when the map D0: R0(X)->-C, given by D0f=f'(x) is norm- continuous, and in this case there is exactly one bounded point derivation at x (up to constant multiples), namely, the extension of D0 to R(X). If x g Xo, then there is exactly one point derivation at x, and it is bounded. For boundary points there may be no derivations, no bounded derivations and infinitely many linearly independent unbounded deriva- tions, or one (up to constant multiples) bounded derivation and infinitely many linearly independent unbounded derivations. These are the only three possibilities. The situation is neatly described in [1], [2]. Necessary and sufficient conditions for the existence of a bounded point derivation at a point x e X have been given in [3]. Browder's derivation theorem [2], and Browder's metric density theorem [1] together imply the following. Theorem 1. The set of points of X at which point derivations of R(X) exist contains no isolated points. Received by the editors October 18, 1972. AMS (MOS) subject classifications(1970). Primary 46J15, 46J10, 30A82, 41A20. Key words and phrases. Bounded point derivation, uniform norm, Swiss Cheese. 1 This work was supported in part by the National Science Foundation, under grant NSF-GP 28574. © American Mathematical Society 1973 559 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use