Undergraduate Mathematics Seminar

Abstract: In a standard calculus course, one learns that the properties of continuity and differentiability are inherently related, i.e. a differentiable real-valued function is continuous. It is also understood that the converse is not necessarily true. This property begs the question, “Can a function be continuous everywhere, but differentiable nowhere?” In fact, the answer to this question is positive, which can be surprising. Nevertheless, Karl Weierstrass constructed his now-famous “Weierstrass function” which exhibits this property. In this seminar, I will discuss the properties of everywhere continuous, nowhere differentiable functions and I will prove the nowhere differentiability of the Weierstrass function. If time remains, continuity will also be discussed.