Infinity does not exist. Nor do uncomputable numbers. No do other infinities.
A real number is a number that is computable in a finite length of time, not an uncountable set of equivalences.
A function is an algorithm for turning some set of numbers into another set in a finite number of steps, not a mapping that needs to be defined at an uncountably infinite number of points.
Mathematicians who disagree with the above are autistic and their ideas have no real world application.