The relationship between LipFα([a,b]) and BVFα,p([a,b])

In 2008/2009 Castillo and Trousselot proposed a new concept of ϑ-Lipschitz continuous function and (p, ϑ)-bounded variation (1 < p < ∞), where ϑ is any strictly increasing and continuous function defined on [a, b]. In this paper, we change the function ϑ with the integral staircase function of order α ∈ (0,1) defined on fractal set F ⊂ [a, b]. The integral staircase function is an increasing continuous function, but it is non-standard differentiable on F. Based on this function, we have a new concept of the fractal Lipschitz continuous function and fractal bounded p-variation. Next, we investigate the relationship between those two functions. In particular, we represent the characterization of the fractal Lipschitz continuous function.