AP Calculus: Continuity – Definition, Types of Discontinuities (Removable, Jump, Infinite)

Continuity – Definition, Types of Discontinuities (Removable, Jump, Infinite)

Concept Summary

• Continuity at a point: A function f is continuous at x = a if f(a) exists, lim? f(x) exists, and lim? f(x) = f(a). This ensures no breaks, jumps, or holes at a.
• Removable discontinuity: A hole in the graph where lim? f(x) exists but f(a)-lim? f(x) (or f(a) is undefined). The discontinuity can be "removed" by redefining f(a).
• Jump discontinuity: The left- and right-hand limits exist but are not equal (lim f(x)-lim f(x)), creating a vertical gap in the graph.
• Infinite discontinuity: The function approaches ±? as x-a (e.g., vertical asymptote), so lim? f(x) does not exist.
• Continuity on an interval: f is continuous on (a, b) if it’s continuous at every point in the interval, and on [a, b] if also continuous from the right at a and left at b.

Core Questions

WHAT (definitional)

Q: What is continuity at a point? A: A function f is continuous at x = a if f(a) is defined, lim? f(x) exists, and both values are equal. Trap/Clarification: Continuity requires all three conditions—missing one (e.g., a hole at a) breaks continuity, even if the limit exists.

Q: What is a removable discontinuity? A: A discontinuity where the limit exists but f(a) is either undefined or unequal to the limit (e.g., a hole in the graph). Trap/Clarification: It’s called "removable" because redefining f(a) to match the limit "fixes" the discontinuity.

WHY (causal/explanatory)

Q: Why is continuity important in calculus? A: Continuity guarantees differentiability (if f is differentiable at a, it must be continuous at a) and enables the use of Intermediate Value Theorem (IVT). Trap/Clarification: Continuity does not imply differentiability (e.g., f(x) = |x| is continuous at x = 0 but not differentiable there).

Q: Why does an infinite discontinuity prevent a limit from existing? A: The limit must be a finite number; if f(x) grows without bound as x-a, the limit is undefined (not ±?). Trap/Clarification: Saying "the limit is ?" is shorthand for "the limit does not exist due to unbounded behavior."

HOW (process/application)

Q: How do you determine if f is continuous at x = a? A: Check three conditions: (1) f(a) exists, (2) lim? f(x) exists, (3) f(a) = lim? f(x). If any fail, f is discontinuous at a. Trap/Clarification: For piecewise functions, check left/right limits separately—if they differ, the limit (and continuity) fails.

Q: How do you classify a discontinuity at x = a? A: Evaluate lim? f(x): - If the limit exists but-f(a)-removable. - If left/right limits exist but differ-jump. - If the limit is ±?-infinite. Trap/Clarification: A hole with a point elsewhere (e.g., f(x) = (x²–1)/(x–1) at x = 1 with f(1) = 5) is still removable (the point is irrelevant to the limit).

CAN (conditions/possibilities)

Q: Can a function be continuous at a point but not differentiable there? A: Yes (e.g., f(x) = |x| at x = 0). Continuity is a weaker condition than differentiability. Trap/Clarification: Differentiability requires continuity, but continuity does not guarantee differentiability.

Q: Under what conditions is a piecewise function continuous at a boundary point x = a? A: The left/right limits must exist and equal each other, and the function value f(a) must match this common limit. Trap/Clarification: The definition of f(a) (which piece it belongs to) doesn’t affect the limit, but it must match the limit for continuity.

Quick Facts & Traps

• Fact: Polynomials, sine/cosine, and exponential functions are continuous everywhere (on ?).
• Trap: "The function has a limit at x = a, so it’s continuous there."-Reality: The limit must also equal f(a).
• Fact: Rational functions (polynomial/polynomial) are continuous wherever the denominator-0.
• Trap: "A hole in the graph means the limit doesn’t exist."-Reality: Holes correspond to removable discontinuities, where the limit does exist.
• Fact: Jump discontinuities often occur in piecewise functions or greatest integer functions (e.g., f(x) = ?x?).
• Trap: "If f is continuous on [a, b], it must be differentiable there."-Reality: Continuity does not imply differentiability (e.g., f(x) = |x| on [-1, 1]).

Rapid-Fire True/False

• Statement: If f is continuous at x = a, then f is differentiable at x = a. Answer: FALSE Why the common mistake happens: Students confuse the necessary condition (differentiability-continuity) with the sufficient condition (continuity does not-differentiability).

• Statement: A function with a vertical asymptote at x = a has a removable discontinuity there. Answer: FALSE Why the common mistake happens: Students mix up infinite discontinuities (asymptotes) with removable ones (holes).

• Statement: If lim? f(x) = L and f(a) is undefined, f has a removable discontinuity at x = a. Answer: TRUE Why the common mistake happens: Students forget that removable discontinuities include cases where f(a) is undefined or mismatched with the limit.