The examples of equivalence classes in my book are mostly related to integers. I would like to know this:
Can a continuous function have equivalence classes?
Sure:
Any function is a relation. Specifically, a function is a relation , such that for any , there is exactly one such that .
Now, for f to be an equivalence relation, its domain has to be its range - .
Specifically, it also has to be reflexive. That is, for each .
What does that tell you about f?