The author studies the set of \(C^r\)-diffeomorphisms of the line with first derivative uniformly bounded away from zero, denoted by \(\mathrm{Diff}^r_0(\mathbb{R})\). One of the main theorems proven is that if \(r =1, 2, \dots , \infty\), and \(\omega\) (real analytic) then there exists a \(C^0\)-open and \(C^r\)-dense set \(U \subseteq\mathrm{Diff}^r_0(\mathbb{R})\) such that the topological entropy is continuous on \(U\) with respect to the strong \(C^0\)-topology. By this result, the topological entropy is locally constant at any uniformly expanding or zero entropy map \(f \in U \cap\mathrm{Diff}^1_0(\mathbb{R})\). The author also provides some examples where the entropy map is not continuous. Additionally, it is shown for \(r =1, 2, \dots , \infty\), and \(\omega\) (real analytic) that there is a generic subset of the set of \(C^r\)-diffeomorphisms of the line where the entropy map is continuous.
These results are analogous to those in a previous paper where the author considered the set of \(C^r\)-diffeomorphisms of the line with bounded first derivative [Discrete Contin. Dyn. Syst. 37, No. 9, 4753–4766 (2017; Zbl 1369.37021)].