Problem. Given a function , show that the following set is at most countable:
My Solution. Denote
then it remains to show that each of 's is countable.
We prove this by showing that each is in fact isolated by an open interval (and hence each point can be identified with a rational number).
Fact. For every , each of points in is isolated.
By isolated we mean for every , there is a such that
In other words, the derived set of , , satisfies .
Proof. Suppose not, i.e., and can't be isolated, i.e., . Then for every , we can find an , such that .
But means that exists with , there will be a with
Since both , as exists, we have , which is absurd.
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