The Morse-Sard theorem is a fundamental result in mathematical analysis, with many applications in differential
topology and geometry. For example, it can be used to prove that the unit sphere is simply connected in
dimensions larger than 1 or that any smooth manifold can be embedded inside an Euclidean space. In it’s
simplest form, it asserts that the set of critical values of a real smooth function is of measure 0. The theorem has
various variants, some of which are entirely self-contained and accessible to third year undergraduates. Despite
the attractive simplicity of the statement and its proof, this theorem is often omitted from the curricula of
courses in multi-variable calculus or measure theory. I will prove particularly simple cases of the theorem, which
would demonstrate that it can be combined in analysis courses at an advanced undergraduate level. Any
student with knowledge of basic point set topology and measure theory would be able to follow the
presentation of the material.