Taking g(x) = f(x) + epsilon*cos(bx) adds to the derivatives a function that bounces between -b*epsilon and b*epsilon while g remains within an epsilon envelope

Taking g(x) = f(x) + epsilon*cos(bx) adds to the derivatives a function that bounces between -b*epsilon and b*epsilon while g remains within an epsilon envelope of f. Take b large enough to fit your needs, e.g. 100/epsilon times an upper bound on |f'|. Add a scaled version of the Weierstrass function instead if you don't want a derivative at all.