We consider an old problem on interpolation at critical points of an algebraic polynomial due to Chandler Davis. We essentially reproduce the proof given by this author in [C. Davis, 4714: Proposed solution: Problem 4653, Amer. Math. Monthly 64(9) 729-730 (1957)]. We provide a detailed account of Davis' proof, adding some details that, in our opinion, make it even easier to follow. Some illustrative examples are also given.