In this paper, we investigate the concept of Abel statistical quasi Cauchy sequences. A real function f is called Abel statistically ward continuous if it preserves Abel statistical quasi Cauchy sequences, where a sequence (αk) of point in R is called Abel statistically quasi Cauchy if limx→1− (1 − x)∑k:|∆αk |>ε xk = 0 for every ε > 0, where ∆αk = αk+1 − αk for every k ∈ N. Some other types of continuities are also studied and interesting results are obtained. It turns out that the set of Abel statistical ward continuous functions is a closed subset of the space of continuous functions