A real function f is continuous if and only if (f(xn)) is a convergent sequence whenever (xn) is convergent and a subset E of R is compact if any sequence x = (xn) of points in E has a convergent subsequence whose limit is in E where R is the set of real numbers. These well known results suggest us to introduce a concept of forward continuity in the sense that a function f is forward continuous if limn›? ?f(xn) = 0 whenever limn›? ?xn = 0 and a concept of forward compactness in the sense that a subset E of R is forward compact if any sequence x = (xn) of points in E has a subsequence z = (zk) = (xnk) of the sequence x such that limk›? ?zk = 0 where ?zk = zk+1-zk. We investigate forward continuity and forward compactness, and prove related theorems. © 2011 by Eudoxus Press,LLC All rights reserved.