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Suppose $f(x):=\left\{\begin{matrix}1&\text{ if }x\in\mathbb Q \\\ 0&\text{ if }x\not\in\mathbb Q.\end{matrix}\right.$ At how many points is $f:\mathbb R\rightarrow\mathbb R$ continuous? Assume the standard topology on $\mathbb R$.

0

1

Countably infinitely many.

Uncountably many.