1.10 Exploring Types of Discontinuities

A discontinuity is removable if the two one-sided limits at the point are equal (so the limit exists) but the function’s value either is missing or doesn’t equal that limit—think a hole you can “fix” by redefining f(a) to be the limit. Algebraically, removable discontinuities often come from a common factor that cancels (e.g., (x−a) in numerator and denominator). It’s nonremovable if either - a jump discontinuity: left-hand limit ≠ right-hand limit (no single limit exists), or - an infinite/essential discontinuity: the limit blows up (vertical asymptote) or oscillates so no finite limit exists. Use the formal continuity definition: f is continuous at a if lim_{x→a} f(x) exists and equals f(a) (CED LIM-2.A). For quick practice and examples (holes vs jumps vs vertical asymptotes) see the Topic 1.10 study guide (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he)—and tackle lots of problems at (https://library.fiveable.me/practice/ap-calculus).

Short answer: a jump discontinuity happens when the left- and right-hand limits at a point exist but are different (finite values)—the graph “jumps” from one y-value to another. A vertical asymptote (an infinite discontinuity) happens when at least one one-sided limit is infinite (±∞)—the function blows up near that x-value. How to tell on a graph or from limits: - Jump: lim(x→a−) f(x) and lim(x→a+) f(x) are both finite but not equal → jump/step discontinuity. - Vertical asymptote: at least one of lim(x→a−) f(x) or lim(x→a+) f(x) = ±∞ → vertical asymptote (infinite discontinuity). Removable vs. these: removable is where both one-sided limits equal the same finite value but f(a) is missing or different. For AP: justify continuity/discontinuity using one-sided limits per LIM-2.A (see the Topic 1.10 study guide for examples) (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he). For extra practice, check unit practice problems (https://library.fiveable.me/practice/ap-calculus).

A hole (removable discontinuity) happens when the two one-sided limits at a point agree (so the limit exists) but the function’s value there is either missing or different. Algebraically you often see this when a factor cancels: f(x) = (x−2)(x+3)/(x−2) has limit 5 at x=2 but maybe f(2) is undefined or set to some other number—that’s a hole. A jump discontinuity happens when the left-hand and right-hand limits are different—lim x→a− f(x) ≠ lim x→a+ f(x). The limit doesn’t exist, so continuity fails in a non-removable way; this is common with piecewise or step functions. (Infinite/vertical-asymptote discontinuities are when one or both one-sided limits blow up to ±∞.) For AP you must use the formal continuity definition and one-sided limits to justify which type it is (CED LIM-2.A). For extra examples and practice, check the Topic 1.10 study guide (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he), the Unit 1 overview (https://library.fiveable.me/ap-calculus/unit-1), and lots of practice problems (https://library.fiveable.me/practice/ap-calculus).

Step-by-step: find where a function is discontinuous 1. Check domain first—any x not in the domain is a discontinuity candidate (e.g., division by 0, even root of negative). 2. For each candidate a, use the formal continuity test (CED LIM-2.A): f is continuous at a iff the left-hand limit, right-hand limit, and f(a) all exist and are equal. So compute lim x→a− f(x), lim x→a+ f(x), and f(a). 3. Classify: - If both one-sided limits equal L but f(a) ≠ L (or f(a) undefined) → removable discontinuity (a hole); often appears when a factor cancels algebraically. Try algebraic simplification/cancel common factors. - If left and right limits exist but are different → jump (step) discontinuity (common in piecewise/step functions). - If at least one one-sided limit is infinite → infinite (essential) discontinuity/vertical asymptote. 4. For rational functions: factor and cancel; zeros of denominator that remain → vertical asymptotes; zeros that cancel → holes. 5. For piecewise functions: check one-sided limits directly at the boundary points. 6. Always justify using one-sided limits and the definition (this is what AP expects—LIM-2.A). For a quick study refresher, see the Topic 1.10 study guide (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he). For mixed practice problems, try the unit practice page (https://library.fiveable.me/practice/ap-calculus).

There are three CED-listed types of discontinuities and how to spot them: 1) Removable (a “hole”): limit as x→a exists (left and right equal) but f(a) is either undefined or not equal to that limit. Algebra tip: a factor cancels (common factor in numerator/denominator). You can “remove” it by defining f(a) to be the limit. 2) Jump (piecewise/step): left-hand and right-hand limits at a are finite but different (lim x→a− f(x) ≠ lim x→a+ f(x)). The overall limit doesn’t exist, and continuity fails because one-sided limits disagree. 3) Infinite / vertical-asymptote (essential/infinite): at least one one-sided limit is ±∞ (or fails to be finite). Graphically the function shoots up/down near x=a—can’t make it continuous by changing f(a). Use the formal continuity-at-a-point test: f is continuous at a iff (i) f(a) defined, (ii) limit exists (both one-sided equal), (iii) limit = f(a). For more examples and AP-style practice, see the Topic 1.10 study guide (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he) and Unit 1 overview (https://library.fiveable.me/ap-calculus/unit-1).

Use the definition whenever you need a rigorous justification that a point is continuous—especially on free-response or when the graph/algebra is ambiguous. The definition: f is continuous at a if and only if (1) f(a) is defined, (2) the limit lim_{x→a} f(x) exists (both one-sided limits equal), and (3) lim_{x→a} f(x) = f(a). How to apply that in practice (CED vocabulary): - If algebra gives a hole (common factor cancels), check the limit and compare to the defined value to spot a removable discontinuity. - If left- and right-hand limits differ, you’ve got a jump (step) discontinuity—the limit doesn’t exist. - If a limit blows up to ±∞ (vertical asymptote), the point is an infinite discontinuity. - At endpoints use one-sided limits (only the side approaching the endpoint matters). On the AP exam you’ll often be asked to justify continuity at a point (LIM-2.A), so write the three parts above and compute the relevant one-sided limits or value. For more worked examples and practice, see the Topic 1.10 study guide (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he), the Unit 1 overview (https://library.fiveable.me/ap-calculus/unit-1), and the AP practice bank (https://library.fiveable.me/practice/ap-calculus).

A removable discontinuity (a “hole”) shows up on a graph when the two one-sided limits at a point exist and are equal, but the function’s value there is either missing or is a different y-value. So when you look: - Check the left-hand and right-hand behavior: if both approach the same finite y, the limit exists. - Then check the dot at that x: if there’s a hole (open circle) or a filled dot at a different y, the discontinuity is removable. - If left ≠ right, it’s a jump discontinuity. If the graph shoots to ±∞, it’s an infinite/vertical-asymptote discontinuity. On the AP, justify this with the definition of continuity: f is continuous at a if lim (x→a) f(x) = f(a) (LIM-2.A). You can review examples and quick checks in the Topic 1.10 study guide (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he). For more practice problems that mirror AP style, try the practice bank (https://library.fiveable.me/practice/ap-calculus).

A function f is continuous at a point a exactly when all three parts of the formal definition hold: 1) f(a) is defined. 2) The limit lim_{x→a} f(x) exists (both one-sided limits equal). 3) lim_{x→a} f(x) = f(a). So the quick “checklist” you can use: Is the value defined? Do the left and right limits match? Does that common limit equal the function value? For endpoints use one-sided limits (e.g., right-hand limit at left endpoint). This is exactly what the AP CED expects you to justify (LIM-2.A). How this helps identify discontinuity types: - Removable (a hole): limit exists, but f(a) is either undefined or =/≠ limit—often fixed by canceling a common factor. - Jump: left and right limits exist but differ. - Infinite/vertical asymptote: at least one one-sided limit is ±∞ (limit does not exist). For practice applying the checklist and spotting types, see the Topic 1.10 study guide (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he) and try problems from the Unit 1 page (https://library.fiveable.me/ap-calculus/unit-1) or the practice set (https://library.fiveable.me/practice/ap-calculus).

Short answer: yes—a vertical asymptote always means the function is discontinuous at that x-value. Why: continuity at a point x = a requires (1) f(a) is defined, (2) the two one-sided limits exist and are equal, and (3) the common limit equals f(a). A vertical asymptote means at least one one-sided limit blows up to ±∞ (or fails to exist in that infinite way), so condition (2) fails. In AP language this is an infinite (essential) discontinuity—listed in the CED as “discontinuities due to vertical asymptotes” / “infinite discontinuity.” Example: f(x)=1/(x−2) has a vertical asymptote at x=2, so f is discontinuous at 2. Contrast: removable discontinuities (holes) happen when a limit exists but f(a) is missing or mis-valued; jump discontinuities happen when left and right limits are finite but unequal. For more on types and one-sided limits, see the Topic 1.10 study guide (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he). For extra practice problems, try the practice library (https://library.fiveable.me/practice/ap-calculus).

Short answer: no—at a single point a function can’t really have two different types of discontinuities at once. The type is determined by the behavior of the one- and two-sided limits at that point, and those behaviors are mutually exclusive. Why: - If the two-sided limit L = lim_{x→a} f(x) exists but f(a) ≠ L, that’s a removable discontinuity (a hole). - If the left and right limits exist but are different (lim_{x→a^-} f ≠ lim_{x→a^+} f), that’s a jump discontinuity. - If one or both one-sided limits blow up to ±∞ (or the function has a vertical asymptote), that’s an infinite discontinuity. Because these definitions depend on whether the two-sided limit exists, whether one-sided limits exist and are equal, or whether limits are infinite, only one category fits any given point. Use the formal continuity definition and one-sided limits to justify your conclusion on the exam (CED LIM-2.A). See the Topic 1.10 study guide for examples and practice: (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he). For extra practice problems, check (https://library.fiveable.me/practice/ap-calculus).

Start by using the definition of continuity at a point: f is continuous at a if lim_{x→a} f(x) exists and equals f(a). To classify a discontinuity on the AP exam, follow these quick steps: 1. Check f(a) (is the point defined?). 2. Compute left- and right-hand limits, lim_{x→a^-} f(x) and lim_{x→a^+} f(x). - If both one-sided limits exist and are equal but ≠ f(a) (or f(a) is undefined) → removable discontinuity (a hole). Often you can show this by algebraic cancellation of a common factor. - If the one-sided limits exist but are unequal → jump discontinuity (common for piecewise/step functions). - If at least one one-sided limit is infinite or doesn’t exist due to blow-up → infinite/essential discontinuity (vertical asymptote). On the exam justify your answer: state the one-sided limits and whether they equal each other and f(a). For worked examples and targeted practice, see the Topic 1.10 study guide (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he) and the Unit 1 overview (https://library.fiveable.me/ap-calculus/unit-1). For more problems, use the practice bank (https://library.fiveable.me/practice/ap-calculus).

A “removable discontinuity” and a “hole” are basically the same phenomenon, but with a small distinction in language: a removable discontinuity is the type of discontinuity (in the CED list) where the limit exists at a point but the function isn’t continuous there; a “hole” is the graph picture you usually see for that discontinuity. Concretely: if f(x) = (x^2 − 1)/(x − 1) for x ≠ 1, then the limit as x→1 exists (equals 2) because the (x−1) cancels—that’s a removable discontinuity. If f(1) is not defined (or is defined as something ≠ 2) the graph has a hole at (1,2). To justify on the AP exam, use the definition of continuity: limit (both one-sided) exists and equals f(a). See the Topic 1.10 study guide for examples and practice (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he). For more practice problems, check the unit page (https://library.fiveable.me/ap-calculus/unit-1) or the practice sets (https://library.fiveable.me/practice/ap-calculus).

When you see a rational function, find discontinuities fast by factoring and checking for cancellations: 1. Factor numerator and denominator completely. 2. Cancel any common factors. - If a factor cancels at x = a (numerator and denominator both zero), you have a removable discontinuity (a hole). The limit exists but f(a) is undefined or redefined. 3. After cancellation, any x where the denominator = 0 gives an infinite discontinuity (vertical asymptote). One-sided limits blow up → not continuous. 4. Jump discontinuities don’t occur for simple rational functions—they show up for piecewise/step functions; detect them by comparing left- and right-hand limits (they’re unequal). 5. Always justify using the definition of continuity: limit exists, equals f(a), and f(a) is defined (CED LIM-2.A). For practice, work a few problems (and check the Topic 1.10 study guide on Fiveable (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he) and more unit review/practice at (https://library.fiveable.me/ap-calculus/unit-1) and (https://library.fiveable.me/practice/ap-calculus)).

A jump discontinuity happens when the left-hand and right-hand limits at a point both exist as finite numbers but aren’t equal—so the two “sides” of the graph jump to different y-values. That’s different from the other types in the CED: - Removable (a “hole”): the two one-sided limits (and the overall limit) exist and are equal, but f(a) is missing or different—you could “fix” it by redefining f(a). - Infinite (vertical asymptote): at least one one-sided limit is infinite (or doesn’t exist because it blows up), so the graph shoots off—not a finite jump. Use the formal continuity definition (LIM-2.A): f is continuous at a if lim x→a− f(x) = lim x→a+ f(x) = f(a). For a jump, the first equality fails because the one-sided limits are unequal. Common example: step or piecewise functions. For a short review, see the Topic 1.10 study guide (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he). For extra practice, try problems at (https://library.fiveable.me/practice/ap-calculus).

Your calculator errors because the function is actually undefined at that x-value—not a calculator bug. AP continuity types explain why: - Removable (a hole): expression like (x^2−1)/(x−1) cancels to x+1 but is undefined at x=1. A calculator will give a “math error” if you try to evaluate exactly at x=1 because the original formula has division by zero. - Jump or piecewise: left- and right-hand values differ, so a single y-value doesn’t exist. - Infinite/vertical asymptote: values blow up, so the calculator may display huge numbers, ERR, or a blank. Workarounds: - Algebraically simplify/cancel factors first, then evaluate the simplified expression to find the removable limit (justify continuity using the definition). - Use one-sided numeric approach: make a table of x-values approaching the point from left and right (trace or TABLE) to estimate limits. - Zoom/resize and increase resolution to see behavior near vertical asymptotes or holes. - For piecewise functions, graph each piece separately or use the calculator’s piecewise/if() feature so it doesn’t try to evaluate outside a piece’s domain. Remember: on the AP exam some parts ban calculators, so you must justify continuity with algebra/one-sided limits (LIM-2.A). For a quick review, check the Topic 1.10 study guide (https://library.fiveable.me/ap-calculus/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he). For extra practice problems, see (https://library.fiveable.me/practice/ap-calculus).