In the 1860's Karl Weirstrass constructed a function that was continuous at every point yet nowhere differentiable. Intuitively we think of continuous functions as smooth, with occasional points that are sharp. Weierstrass's function, "sharp" everywhere, confounded mathematicians, and was regarded as pathological. Even more amazing is the finding that continuous nowhere differentiable functions are by far more common than continuous differentiable functions. This paper presents some examples of continuous nowhere differentiable functions and proofs of their peculiar property. These functions exemplify how rigor in mathematics may lead to counter-intuitive results, an important lesson for all students of mathematics.
|Presenter:||Michelle Beshty (Undergraduate Student)|
|Time:||9:55 am (Session I)|