How can I go about showing that if I have a function $f(x)$ that is continuous on an interval $I$, and there is a set $S$ where $f(x)$ fails to be 1 to 1. I need to show that $S$ is either uncountable or empty.
What is known:
$f(x) = f(y)$ for $y$ $\neq$ $x$ and $y$ $\in$ $I$ is the definition of one to one.
I know that if there exists an interval where 1-1 fails, I can use the proof that the subset of the reals is uncountable. However, what about singletons? Or the empty set?