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 $\displaystyle f:A \to B$ is a relation $\displaystyle f \subseteq A \times B$, such that for any $\displaystyle a \in A$, there is exactly one $\displaystyle b \in B$ such that $\displaystyle (a,b) \in f$.
Now, for f to be an equivalence relation, its domain has to be its range - $\displaystyle f: A \to A$.
Specifically, it also has to be reflexive. That is, for each $\displaystyle a \in A, \ (a,a) \in f$.
What does that tell you about f?
Well, what I meant was that you can get the result without assuming continuity, but the function you get (identity function) is always continuous. The equivalence class of $\displaystyle x \in \mathbb{R}$ is $\displaystyle [x] = \{ y \in \mathbb{R} : (x,y) \in f \}$ $\displaystyle = \{ y \in \mathbb{R} : f(x)=y \} = \{ y \in \mathbb{R} : x = y \} = \{ x \}$
So each point's equivalence class is itself.