Point of Discontinuity

A point of discontinuity (or hole) happens when a rational function simplifies algebraically to a cleaner expression, but the original, unsimplified function is still undefined at one particular x-value. The graph looks like a normal continuous curve, except for a single missing point.

f(x) = (x² − 4)/(x − 2) looks undefined at x = 2, since the denominator becomes zero there. But the numerator factors: x² − 4 = (x−2)(x+2), so f(x) = (x−2)(x+2)/(x−2) = x + 2, for every x except 2. The graph is the line y = x + 2, with a hole at the point (2, 4) — the y-value the line would have there, if the function were defined.

Factor both the numerator and denominator of the rational function. If a factor appears in both and cancels, the x-value that makes that factor zero is a removable discontinuity — a hole, not a true vertical asymptote. To find the hole's y-coordinate, substitute that x-value into the simplified expression (never the original).

The distinction matters: if the factor causing the zero cancels, you get a single-point hole. If it doesn't cancel, the function grows without bound near that x-value instead — a vertical asymptote. Studying the full graph of a rational function means checking every restricted value for which case applies.

Related lessons

• Algebra