$f(x) = x$ when x is rational, $f(x) = 0$ when x is irrational. Find all points at which $f$ is continous.

real-analysis continuity proof-verification

Solution 1:

This proof looks great. Using the sequential characterization of continuity to show discontinuity at every non-zero point, and then the $\epsilon-\delta$ characterization to show continuity at zero is smart. Good work. :)

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