Noncontinuity and an induced equivalence relation Can someone give me an example of a map which is not continuous such that $f(\{a\}) = f(\{b \})$ induces an equivalence relation $ \{ a \} \sim \{ b \} $?

Let $f:\{a,b,c,\}\to\{0,1\}$.  Let $f(a)=f(b)=0$ and $f(c)=1$.  Let $x\sim y$ precisely if $f(x)=f(y)$.  Then we have
\begin{align}
a & \sim a \\
a & \sim b \\
a & \not\sim c \\  \\
b & \sim a \\
b & \sim b \\
b & \not\sim c \\  \\
c & \not\sim a \\
c & \not\sim b \\
c & \sim c
\end{align}
This is an equivalence relation on the set $\{a,b,c\}$ with two equivalence classes: $\{a,b\}$ and $\{c\}$.
