A new class of generalized open sets in a topological space, called e-open sets, is introduced and some properties are obtained by Ekici [6]. This class is contained in the class of $delta$-semi-preopen (or $delta-eta$-open) sets and weaker than both $delta$-semiopen sets and $delta$-preopen sets. In order to investigate some different properties we introduce two strong form of e-open sets called e-regular sets and e-$ heta$-open sets. By means of e-$ heta$-open sets we also introduce a new class of functions called strongly $ heta$-e-continuous functions which is a generalization of $ heta$-precontinuous functions. Some characterizations concerning strongly $ heta$-e-continuous functions are obtained.