Summer of Math Exposition

Based on the Fundamental Theorem of Calculus, you might assume that every derivative is integrable. Shockingly, this is not actually true! In this video, I show you the construction of a function which has a derivative which is not integrable, delivering a counterexample to this wrongful assumption. The video consists of three parts: first up is a crucial lemma for proving nonintegrability, second is the construction of a set on which the derivative will be discontinuous, and last is the construction of the function and its nonintegrable derivative. The video is aimed at undergraduates who have completed their first year of mathematics, specifically at people who have taken a course in Real Analysis.