It is in the field of numerical analysis that this "easy-looking" function, also known as the Runge function, exhibits a behavior so idiosyncratic that it is mentioned even in most undergraduate textbooks. In spite of the fact that the function is infinitely differentiable, the common procedure of (uniformly) interpolating it with polynomials that use equally spaced points in intervals of suitable length centered at zero has been proven to be an impossible task. In this article, the authors present a proof for the reasons behind the failure of the usual polynomial interpolation of the Runge function which is complete and accessible to undergraduate students with some background in real and numerical analysis. The main idea of the proof is from Isaacson and Keller's work and its presentation is in a self-contained form which is broken down in detailed and easy-to-follow steps. (Contains 4 figures.)