# How do you know a limit does not exist?

Most limits DNE when ${\lim}_{x \to {a}^{-}} f \left(x\right) \ne {\lim}_{x \to {a}^{+}} f \left(x\right)$, that is, the left-side limit does not match the right-side limit. This typically occurs in piecewise or step functions (such as round, floor, and ceiling).
So, an example of a function that doesn't have any limits anywhere is f(x) = {x=1, x in QQ; x=0, otherwise}. This function is not continuous because we can always find an irrational number between 2 rational numbers and vice versa.