We define continuity rigorously and show that continuous functions satisfy strong extensionality. Continuity, discontinuity, and differentiability are discussed. We find that classical gems like the Intermediate Value Theorem and Brouwer’s Fixed Point Theorem are not constructively valid, as revealed by weak counterexamples. However, these theorems can be salvaged by proving approximate versions. In particular, using Sperner’s Lemma, we do this for the Fixed Point Theorem in one and two dimensions.