We often come across functions like f(x)= x, x belongs to rational and -x, when x belongs to irrational, which are continuous at just one (or more isolated) point, in this example at x=0. I googled to look at the graph of such a function, with no luck. Even desmos wouldn't let me type such a function. So, does anyone know how to find graph of such a function, can anybody share it here?
2026-05-14 08:49:59.1778748599
Graph of a single point continuous function.
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One can imagine what the graph would look like: since the rationals and irrationals are both dense, the graph would appear to be an X shape: the union of both the lines $f(x)=x$ and $f(x)=-x$. Of course, if one could "zoom in infinitely", one would see that the lines are not really lines, but points that are infinitesimally close.