Limits and Continuity

To check continuity at a point and classify any discontinuity you find, you need a three-step test. Many students confuse the limit of a function with its value at the point, but limits and continuity require you to compare both. Here is the exact procedure from the AP Calculus AB/BC Course and Exam Description, Unit 1 (effective Fall 2020).

A function f(x) is continuous at x = a if and only if all three of these hold:

1. f(a) is defined. The function has a real value at x = a. If the point is not in the domain, condition 1 fails immediately.
2. lim_x→a f(x) exists. The left-hand limit and right-hand limit must both exist and be equal. If they are not, the limit does not exist (DNE) and condition 2 fails.
3. lim_x→a f(x) = f(a). The limit and the function value match.

If any condition fails, the function has a discontinuity at x = a. Use the type of failure to classify the discontinuity.

The continuity definition is simple: a continuous function has no breaks, jumps, or holes. You can draw its graph without lifting your pencil. But the formal definition matters for classification. A function is continuous on an interval if it is continuous at every point in that interval. The AP Calculus AB/BC CED Unit 1 (10-12% of the AB exam, 4-7% of the BC exam) requires you to apply this definition to classify discontinuities.

When a function fails the three-condition test, the failure pattern tells you which of the three types of discontinuity it is. Each type corresponds to a specific failure in conditions 2 or 3.

Discontinuity Type	Condition That Fails	Limit Behavior	Graph Appearance
Removable Discontinuity	Condition 3 only (limit exists, but f(a) is undefined or mismatched)	Limit exists as a finite number	Hole in the graph
Jump Discontinuity	Condition 2 (left and right limits exist but are not equal)	Left and right limits are finite but different values	Break; graph jumps from one y-value to another
Infinite Discontinuity	Condition 2 (at least one one-sided limit is infinite)	Limit is infinite (DNE as a finite number)	Vertical asymptote

A removable discontinuity occurs when the limit exists but the function is not defined at the point, or its value differs from the limit. The term 'removable' comes from the fact that you could redefine f(a) to equal the limit, and the function would become continuous at that point.

Example: Factor and Cancel

Example: f(x) = (x², 1)/(x, 1). At x = 1, f(1) is undefined (division by zero). The limit as x → 1 is 2. This is a removable discontinuity. You can factor and cancel: (x, 1)(x + 1)/(x, 1) = x + 1, so the limit is 2. The graph has a hole at (1,2).

Failure case: Students often cancel the factor (x, 1) without noting that it is zero at the point. This cancels the hole but does not erase the discontinuity. Direct substitution fails here because the function is not continuous at x = 1.

A jump discontinuity occurs when the left-hand limit and right-hand limit both exist as finite numbers but are not equal. The one-sided limits are different, so the two-sided limit does not exist. The graph makes a sudden jump from one y-value to another.

Example: Piecewise Function

Example: A piecewise function where f(x) = x + 1 for x < 2 and f(x) = 3x, 1 for x ≥ 2. The left-hand limit at x = 2 is 3; the right-hand limit is 5. They are different, so the limit does not exist. The function is not continuous at x = 2, and the discontinuity is a jump.

Jump discontinuities are common in piecewise functions where the pieces meet at different y-values. They cannot be 'removed' by redefining a single point.

An infinite discontinuity occurs when at least one one-sided limit is infinite (∞ or, ∞). The function grows without bound as x approaches the point. This creates a vertical asymptote. Common examples include rational functions where the denominator is zero and the numerator is non-zero.

Example: Rational Function

Example: f(x) = 1/(x, 3)². As x → 3 from either side, the denominator approaches 0, so the function values become arbitrarily large. The limit is infinite, so the limit does not exist as a finite number, but the AP Calculus CED convention allows writing lim_x→3 1/(x, 3)² = ∞. The graph has a vertical asymptote at x = 3.

Failure case: Do not confuse infinite discontinuity with a limit that does not exist because of oscillation (like sin(1/x) at 0). In oscillation, no single value or infinite direction is approached. Here, the function clearly goes to ±∞.

A common AP Calculus AB/BC problem: Find the value of k that makes a piecewise function continuous at the boundary point. The method is to set the left-hand limit equal to the right-hand limit, and also equal to the function's value at that point.

Example: Solve for k

Let f(x) = 2x + k for x < 1, and f(x) = 3x² for x ≥ 1. For continuity at x = 1, three things must match:

• Left-hand limit: lim_x→1^, (2x + k) = 2(1) + k = 2 + k.
• Right-hand limit: lim_x→1⁺ 3x² = 3(1) = 3.
• Function value: f(1) = 3(1)² = 3.

Set the left-hand limit equal to the right-hand limit: 2 + k = 3, so k = 1. Check that f(1) equals the limit: f(1) = 3, and the limit from the left with k = 1 is 2 + 1 = 3. All three conditions are satisfied, so the function is continuous at x = 1 when k = 1.

Failure case: Do not forget to check both one-sided limits. If the left-hand limit and right-hand limit are not equal, no value of k can fix the jump discontinuity.

The Intermediate Value Theorem (IVT) states: If f is continuous on the closed interval [a, b] and L is any number between f(a) and f(b), then there exists at least one c in [a, b] such that f(c) = L. This theorem relies entirely on continuity. If the function has a jump or infinite discontinuity on the interval, the IVT does not apply. Use it to prove that a root exists without finding it.

The IVT is a separate topic from the Squeeze Theorem and L'Hôpital's Rule, which address limit evaluation rather than continuity. For epsilon-delta proofs, see a dedicated real analysis text like Spivak's 'Calculus', ch. 5.

If direct substitution gives an indeterminate form like 0/0, the function may have a removable discontinuity. Factor, rationalize, or cancel to find the limit. If the left and right limits differ, you have a jump discontinuity, and the limit does not exist. If the function goes to ±∞, you have an infinite discontinuity. The single thing that most often goes wrong is forgetting to check whether the limit exists at all before comparing it to the function value. Always test condition 2 before condition 3.