A Note on the Continuous and Nowhere Differentiable Function

We continue the studies of derivatives of the non-harmonic series ∑cnexp(iλnx). We consider replacing sinλnx with some step function fn:[0,1)→{0,1,2}. The purpose of this paper is to show that f(x)=∑fn(x)2n is continuous and nowhere differentiable in (0,1). All functions fn, used to construct f, are created from symmetrical blocks of type M: [0,1,0] and blocks of type L: [0,1,2] and R: [2,1,0], located symmetrically with respect to M. The function f1 takes the value zero in the intervals [0,1/3), [2/3,1) and the value one in the interval [1/3,2/3), i.e., it consists of the block of type M: [0,1,0]. Function f2 takes the value two in the intervals [2/9,3/9) and [6/9,7/9); the value one in the intervals [1/9,2/9), [4/9,5/9), and [7/9,8/9); and the value zero in the remaining intervals. This means a composition of blocks of [0,1,2][0,1,0][2,1,0], i.e., LMR blocks. The function f3 is a symmetric composition of blocks, LLMLMRMRR. These blocks are discontinuous analogs of the functions that produces Schoenberg curves.