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Let $n$ be a fixed positive integer. For any real number $x$, if for some integer $q$, $x=qn+r, \: \: \: 0 \leq r < n$, then we define $x \text{ mod } n=r$. Specify the points of discontinuity of the function $f(x)=x \text{ mod } 3$ with proper reasoning.

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