I know that Dirichlet function has uncountable many discontinuities. I think they are removable, because the discontinuities can be removed by redefining the function values of the rational numbers as 0.
So Dirichlet function is a function that has uncountable many removable discontinuities, then my question is can we construct a function with uncountable many jump discontinuities? If not, how do we prove it is impossible? Thank you.
An odd but similar question is can we have a function that has uncountable many infinite discontinuities?