In this paper the following question is investigated. Given a natural number rr and numbers αj,βj for j = 0,1,…,r satisfying α0 < α1 < ⋯ < αr and ∑j=0rβjαkj={0r!if k=0,1,…,r−1if k=r , is there a 2π-periodic, r − 1 times continuously differentiable function ff such that lim suph↗0h−r(∑j=0rβjf(x+αjh))=lim suph↘0h−r(∑j=0rβjf(x+αjh))=∞, lim infh↗0h−r(∑j=0rβjf(x+αjh))=lim infh↘0h−r(∑j=0rβjf(x+αjh))=−∞ for every x ∈ ℝ? [ABSTRACT FROM AUTHOR]