Weierstrass-like functions on local fields and their p-adic derivatives

The concept of derivatives of functions plays a key role in the study of local fields. Such a definition was given by virtue of pseudo-differential operators by Su in 1992 [Su W. Pseudo-differential operators and derivatives on locally compact Vilenkin groups. Sci China (series A) 1992;35(7A):826–36; Su W. Gibbs–Butzer derivatives and the applications. Numer Funct Anal Optimiz 1995;16(5–6):805–24]. In this paper, a kind of Weierstrass-like functions in the p-series local fields are found, these Weierstrass-like functions [Falconer KJ. Fractal geometry: mathematical foundations and applications. New York: John Wiley & Sons, Inc.; 1990. [1]] are continuous, and m order differentiable with m < 1 but not one order differentiable at any point in its domain.