Jumps of entropy for $C^r$ interval maps

We study the jump of topological entropy for $C^r$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f \in C^r([0; 1])$with$h_{top}(f) > \frac{\log^+ |f’|_\infty}{r}$. To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^r$ interval maps.