Confirming continuity over an interval — AP Calculus BC
1. Definition of Continuity Over an Interval ★★☆☆☆ ⏱ 3 min
A function is continuous over an interval if every point in the interval satisfies all three conditions for continuity at a point: $f(c)$ is defined, $\lim_{x \to c} f(x)$ exists, and $\lim_{x \to c} f(x) = f(c)$. This topic accounts for 3-6% of your total AP Calculus BC exam score, and acts as a foundational justification step for core concepts like the Intermediate Value Theorem, differentiability, and integration.
2. Continuity of Elementary Functions Over Open Intervals ★★☆☆☆ ⏱ 4 min
Elementary functions (polynomials, rational functions, trigonometric functions, exponentials, logarithms, and root functions) have a key simplifying property: every elementary function is continuous at all points in its domain. This result comes directly from limit laws, so you do not need to check every point individually in an interval.
To confirm continuity of an elementary function over an open interval, you only need to verify that the entire interval is contained within the function's domain. If there are no excluded points (division by zero, negative radicands, non-positive log arguments) inside the interval, the function is continuous over the full interval.
3. Continuity on Closed Intervals with Endpoints ★★★☆☆ ⏱ 4 min
For closed intervals $[a,b]$, two-sided limits do not exist at the endpoints because the function is only considered on the interval itself. The full definition of continuity on $[a,b]$ requires three conditions:
- $f$ is continuous at every point in the open interval $(a,b)$
- $f$ is right-continuous at the left endpoint $a$: $\lim_{x \to a^+} f(x) = f(a)$
- $f$ is left-continuous at the right endpoint $b$: $\lim_{x \to b^-} f(x) = f(b)$
This requirement is explicitly tested on AP free-response questions, most often when justifying the use of the Intermediate Value Theorem. Skipping the one-sided endpoint check will cost you justification points.
4. Continuity of Piecewise-Defined Functions Over an Interval ★★★★☆ ⏱ 5 min
Piecewise functions use different expressions for different sub-intervals, so the only possible points of discontinuity inside an interval are the breakpoints (points where the expression changes). To confirm continuity over an interval containing breakpoints, follow these steps:
- Confirm each individual piece is continuous on its open sub-interval (almost always, each piece is elementary, so only check domain per piece)
- Check all three continuity conditions at every breakpoint inside the interval: evaluate the left limit with the left piece, right limit with the right piece, and confirm both equal $f(c)$ at the breakpoint
- If the interval is closed, confirm one-sided continuity at the full interval endpoints as usual
Common Pitfalls
Why: Students confuse the simplified form with the original function's domain; any point not in the original domain is a discontinuity regardless of cancellation.
Why: Two-sided limits do not exist at endpoints when we only consider the function on the interval.
Why: Students assume the function uses the same expression everywhere near the breakpoint, instead of switching expressions at the break.
Why: Students forget that $x=0$ is not in the domain of $\ln(x)$, so the endpoint condition fails.
Why: Students check discontinuities of the outer function regardless of the output of the inner function over the interval.