No doubt pedantic mathematicians had existed before, but what these examples did is actually convince mathematicians in general to be far more concerned about t

No doubt pedantic mathematicians had existed before, but what these examples did is actually convince mathematicians in general to be far more concerned about the subjects of mathematical foundations. E g. an actually consistent and encompassing definition of what a "function" is, is something surprisingly recent. Certainly Euler, Gauß, Leibnitz or Newton did not have one. Only with Cantor's project of set theory did we get something remotely satisfying. Cantor faced significant backlash, which surely would have been much more effective if there weren't examples where actual foundational questions became questions of practical mathematics. And whether Weierstrass had actually found a "function" was one of these questions.