Vol. 5 No 10 - Pi Mu Epsilon

PARTIAL DIFFERENTIATION ON A METRIC SPACE 1by RoAeann t:\o^L&UoSvton HaU. U~uvvui.UyThe "metric differentiability" of functions between abstract metricspaces is a relatively new and undeveloped field of mathematics.metric derivative of a function with real domain was first defined in1935 by W. A. Wilson (See [2]). In 1971, E. Braude arrived at the samedefinition after having researched the topic independently.TheUnlike hispredecessor, Braude sought to develop his concepts of metric differenti-ability for functions with abstract metric domains,In this paper weexplore the idea of partial differentiation for functions defined on thecartesian product of two metric spaces.PaJvtioJi cii.f.f.e~~e.ivU.a^i.ona mvUu.c ApaceLet f be a function from a metric space (X,s) into a metric space(Y,t) and let bX. f is said to be metrically differentiable at b ifb is not discrete and if a real number f(b) exists with the propertythat for every E > 0, a positive number 5 can be found such that ifs(xt, b) < 6, s(xV, b) < 6 and x' # x", thenThe assertion "f: X -* Y is metrically differentiable" means that f ismetrically differentiable at every point of X. (See [2].)Proceeding almost directly from the definition of the metric deriv-ative is the definition of the metric partial derivative.Let f be afunction from a metric space (XxY,p) into a metric space (Z,p ') and let(x,~) be an element of XxY for which every open ball containing (x,y)contains an element (xl,y) with x' # x; f has a metric partial deriva-t ewith respect to x at (x,y) if a rea-1 number /(x,y)exists with theproperty that for every n > 0, a positive number 5 can be found such thatif-p(xl, x) < 6, p(xr', x) < 6 and xt # xu, then'one of several papers produced as a result of a research project fundedby the Research Corporation under the Cottrell College Science Program,grant number c-205/308, directed by Professor Em 3. Braude, Seton HallUniversity.-.I^ pf(f(x', Y), W,IY) - y )