Calculus AB · Unit 1: Limits and Continuity · 14 min read · Updated 2026-05-10
Confirming continuity over an interval — AP Calculus AB
AP Calculus AB · Unit 1: Limits and Continuity · 14 min read
1. Core Definition of Continuity Over an Interval★★☆☆☆⏱ 3 min
Confirming continuity over an interval means verifying a function meets the formal continuity definition at every point in the interval. This topic makes up 10–12% of Unit 1 exam weight, and is a required first step for justifying major theorems like the Intermediate Value Theorem (IVT) and Extreme Value Theorem (EVT) on AP FRQs.
2. Continuity on Open Intervals★★☆☆☆⏱ 4 min
An open interval $(a, b)$ does not include its endpoints, so you only need to verify that $f$ is continuous at every interior point $c$ between $a$ and $b$. Recall the three conditions for continuity at any interior point:
$f(c)$ is defined
$\lim_{x \to c} f(x)$ exists
$\lim_{x \to c} f(x) = f(c)$
A key AP exam shortcut: all elementary functions (polynomials, rational, root, exponential, logarithmic, trigonometric) are continuous at every point in their domain. This means you do not need to test every point individually: you only need to confirm that the entire open interval lies within the function's domain. For piecewise functions, the only possible points of discontinuity inside an open interval are the breakpoints where the function's definition changes.
3. Continuity on Closed Intervals★★★☆☆⏱ 4 min
A closed interval $[a, b]$ includes both endpoints, so the definition for continuity adds two extra one-sided continuity requirements beyond continuity on the interior $(a, b)$. Since the function is only defined for $x \in [a, b]$, we cannot approach the left endpoint $a$ from the left, so we only require right-continuity at $a$, and left-continuity at the right endpoint $b$.
4. Continuity of Piecewise Functions Over Intervals★★★★☆⏱ 3 min
Piecewise functions have different definitions for different subintervals, so they require a targeted check for discontinuities. The only possible points of discontinuity are: (1) breakpoints where the function's definition changes that lie inside your target interval, and (2) points inside any subinterval where the individual piece has a discontinuity (such as a zero denominator).
For any interval that includes a breakpoint inside the interval, you must test all three continuity conditions at that breakpoint by calculating one-sided limits from each side. If the breakpoint is an endpoint of your overall interval, you only check the one-sided limit from inside the interval.
Common Pitfalls
Why: Students memorize continuity rules for interior points and ignore the extra endpoint requirement that is mandatory for theorem justifications.
Why: Students confuse the existence of the limit with the full definition of continuity, which requires the function to be defined at the point.
Why: Students assume that because each piece is continuous on its own, the combined function is continuous over the whole interval.
Why: Students default to two-sided limits out of habit, even though the function is not defined outside the interval.
Why: Students think only interior discontinuities count, but any point in the interval where the function is discontinuous breaks continuity over the whole interval.