In the 19th century, German mathematicians, including Karl Weierstrass, set about challenging assumptions and beliefs in mathematics to put number theory on a stronger theoretical footing, and one of the concepts they focussed on was calculus.
Weierstrass created a function that, according to existing assumptions, should not have been possible because it was continuous everywhere but differentiable nowhere, meaning it violated assumptions that mathematician André-Marie Ampère had made around 1806 that there were only finite points where a derivative did not exist on a continuous function.
The function Weierstrass created was pathalogical in that it became more jagged with closer inspection and had many flavourful effects, making it seemingly useless in a pure mathematical sense, but it has since been used to model phenomena such as Brownian motion and financial markets.
The redefinition of calculus that was required to account for Weierstrass’ function has since grown into the field of analysis.