The pp-adic differentiability of a class of Weierstrass type functions in local fields

This paper will investigate the Weierstrass type function f(x)=∑n=0+∞p−ng(β−nx) where gg is a periodic function on the pp-adic or pp-series local field KpKp. We will prove that if gg lives in the C1,εC1,ε class, then ff is either 11-order pp-adic differentiable at any point or 11-order pp-adic non-differentiable almost everywhere. Moreover, we show that the previous case is an exceptional case.