The paper deals with the question for which piecewise monotone interval maps topological entropy can jump up under small perturbations preserving the number of pieces of monotonicity. It turns out that for continuous transitive maps jumps cannot occur if the number of pieces of monotonicity is smaller than 6, while they can occur if this number is 6 or more. Additionally, unified and simple proofs of the fact that such jumps are impossible for unimodal and Lorenz-like maps of positive entropy are presented.