Exploring Types of Discontinuities

Continuity at a point needs three things: the function is defined there, the two-sided limit exists there, and the limit equals the function's value. Each type of discontinuity fails a different one of these. A removable discontinuity is a pinhole — the limit exists and is finite, but either the function isn't defined there or its value doesn't match the limit. You could patch it by defining the function to equal that limit. A jump discontinuity happens when the left-hand and right-hand limits both exist but are different finite numbers — the graph steps from one height to another, and no single point value can fix that gap. An infinite discontinuity is a cliff that never stops falling or rising — the function heads toward a vertical asymptote, so there's no finite limit to patch toward at all.

For rational functions, factoring tells you which kind you have. If a factor cancels between numerator and denominator, the discontinuity at that x-value is removable — the cancellation is exactly what makes the limit exist even though plugging in gives 0/0. If a factor in the denominator does not cancel, the function grows without bound near that x-value, giving an infinite discontinuity. Jump discontinuities usually show up in piecewise functions rather than rational ones.

This topic sorts every place a function fails to be continuous into one of these buckets, setting up derivatives: a function can only be differentiable where it's continuous, and knowing which kind of break you have tells you exactly why smoothness fails there.