Functions with Closed Graphs

This paper concerns i t s e l f mainly with those functions from one topological or metric.space to another that have closed graphs i n the product space. Their r e l a t i o n s h i p to closed, l o c a l l y closed, compact, continuous and subcontinuous functions i s studied i n order to determine the r e l a t i v e strength of the closed graph condition. The paper c o l l e c t s and i n some cases extends re s u l t s found i n papers by R. V. F u l l e r [ 2 ] , P. E. Long [7] P. Kostyrko ' a n d T. Shalat [ 4 ] , [5] and [ 6 ] . The main theorems deal with; l ) the characterization of continuous functions i n terms of subcontinuity and the closed graph property; 2) a proof that i f f has a closed graph then f i s the l i m i t of a sequence of continuous functions; and 3) a study of the operations under which the class of functions with closed graphs i s closed.