AP® Calculus BC · Unit 1 · Topic 1.10
Exploring Types of Discontinuities
A function breaks in three distinct ways: a removable hole, a jump, or an infinite blow-up, and one-sided limits tell you which one you're looking at.
A function is continuous at x = c when the limit as x approaches c actually equals f(c). When that fails, the failure has a shape. Check the left-hand and right-hand limits separately. If both exist and agree with each other but not with f(c), or f(c) is missing entirely, you have a removable discontinuity: a single hole that algebra created, usually from a factor that cancels. If the left and right limits both exist but disagree with each other, the graph jumps from one height to another, a jump discontinuity, common in piecewise functions where the pieces don't meet. If either one-sided limit heads to infinity, the function has an infinite discontinuity, tied to a vertical asymptote.
The mechanism is always the same question asked twice: what does f(x) do as x sneaks up from the left, and what does it do sneaking in from the right? Removable discontinuities are the only ones you can patch, because a two-sided limit already exists there waiting to be assigned as the function's value. Jumps and infinite breaks have no such limit to hand you, so no single redefinition can close them.
This topic sets up the formal definition of continuity and the Intermediate Value Theorem right after it. Both lean on knowing exactly which kind of break you're dealing with, since IVT only applies on intervals where the function is genuinely continuous, holes included or excluded on purpose.
Key terms
- Removable discontinuity
- A hole where the two-sided limit exists but does not equal f(c), or f(c) is undefined.
- Jump discontinuity
- A break where the left-hand and right-hand limits both exist but are not equal to each other.
- Infinite discontinuity
- A break where at least one one-sided limit is infinite, usually marking a vertical asymptote.
- One-sided limit
- The value a function approaches as x approaches c from only the left or only the right.
Watch it explained
- How it worksHow do one-sided limits separate a removable hole from a jump from an infinite blow-up?
- Worked exampleFind and classify every discontinuity of (x2 - 1)/(x2 - x).
- Worked exampleFind and classify every discontinuity of the piecewise function x + 1 when x < 2 and x2 when x ≥ 2.
- Common mistakeWhy isn't every break in a curve fixable by redefining a single point?
- Ties togetherHow does spotting a removable discontinuity connect to patching it by redefining the function there?
Check yourself
End of topic.