If f is a real‐valued function defined on an interval I, we find that the set {x0ϵI|f(x0‐) is infinite f(x0+) is infinite} (the set of vertical asymptotes of f) is countable. We are then able to conclude that the only set of discontinuities of f which is not necessarily countable is the set {x0ϵI|f(x0‐) does not exist (but is not infinite) and f(x0+)does not exist (but is not infinite)}.