Fun with Continuity

Most functions we see in calculus are continuous everywhere or at all but a few points that can be easily identified. But consider the Dirichlet function:

Q\left( x \right)=\left\{ \begin{matrix} 1 & x\in \ rational\ numbers \\ 0 & x\in \ irrational\ numbers \\ \end{matrix} \right.

Since there is one (actually many) rational numbers between any two irrational number and one (many again) irrational numbers between any two rational numbers, this function is not continuous anywhere!

But a very similar function is continuous at exactly one point (1, 1):

f\left( x \right)=\left\{ \begin{matrix} 1 & x\in \ rational\ numbers \\ x & x\in \ irrational\ numbers \\ \end{matrix} \right.

Can you use this idea to make a function that is continuous at exactly two points?


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