What conditions need to be satisfied for a continuous function to be:
1) A well-defined function
2) An equivalence relation?
Answer those and you will get your result; note that you're not even considering the continuity.
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?