Summary: A concept related to total variation termed \(\mathcal H_1\) condition was recently proposed to characterize the chaotic behavior of an interval map \(f\) by [G. Chen, T. Huang, and Y. Huang [Int. J. Bifuration Chaos Appl. Sci. Eng. 14, No. 7, 2161–2186 (2004; Zbl 1077.37510)]. In this paper, we establish connections between \(\mathcal H_1\) condition, sensitivity and topological entropy for interval maps. First, we introduce a notion of restrictiveness of a piecewise-monotone continuous interval map. We then prove that \(\mathcal H_1\) condition of a piecewise-monotone continuous map implies the non-restrictiveness of the map. In addition, we also show that either \(\mathcal H_1\) condition or sensitivity then gives the positivity of the topological entropy of \(f\).