This study introduced and investigated μ-β ̂g-continuous, almost μ-β ̂g-continuous and weakly μ-β ̂g-continuous functions in generalized topological spaces. Properties, characterizations and relationships among these functions are also considered. Hereafter, it has been proven that a μ-continuous function is μ-β ̂g-continuous. Moreover, a μ-β ̂g-continuous function is almost μ-β ̂g-continuous and an almost μ-β ̂g-continuous function is a weakly μ-β ̂g-continuous function