A function is continuous at a point when its graph can be drawn through that point without a break, hole, jump, or vertical blow-up. Discontinuities matter because they show where a formula changes behavior or where a model stops making physical or mathematical sense. In calculus, continuity is also a key condition for theorems about limits, derivatives, and integrals.
Learning to recognize the main types helps you read graphs and analyze piecewise functions correctly.
At a point x = a, continuity requires three things: f(a) is defined, lim x -> a f(x) exists, and the limit equals the function value. A removable discontinuity happens when the two-sided limit exists but the function value is missing or placed at the wrong height. A jump discontinuity happens when the left-hand and right-hand limits are finite but unequal.
An infinite discontinuity happens when function values grow without bound near x = a, often because of a vertical asymptote.
Key Facts
- Continuity at x = a requires f(a) is defined, lim x -> a f(x) exists, and lim x -> a f(x) = f(a).
- Two-sided limit exists only when lim x -> a- f(x) = lim x -> a+ f(x).
- Removable discontinuity: lim x -> a f(x) exists, but f(a) is undefined or f(a) != lim x -> a f(x).
- Jump discontinuity: lim x -> a- f(x) and lim x -> a+ f(x) are finite but not equal.
- Infinite discontinuity: f(x) -> infinity or f(x) -> -infinity as x approaches a from at least one side.
- A rational function often has a removable discontinuity when a factor cancels and an infinite discontinuity when a noncanceling denominator factor equals zero.
Vocabulary
- Continuity
- Continuity at a point means the function value matches the limit of the function as x approaches that point.
- Removable discontinuity
- A removable discontinuity is a hole or misplaced point where the limit exists but the function is not defined correctly at that x-value.
- Jump discontinuity
- A jump discontinuity occurs when the left-hand and right-hand limits are different finite numbers.
- Infinite discontinuity
- An infinite discontinuity occurs when the function grows without bound near a point, usually creating a vertical asymptote.
- One-sided limit
- A one-sided limit describes the value a function approaches from only the left side or only the right side of a point.
Common Mistakes to Avoid
- Ignoring whether f(a) is defined. A limit can exist at x = a even if the function has a hole there, so you must check the actual function value separately.
- Assuming a graph is continuous because both sides look close. For continuity, the left-hand limit, right-hand limit, and f(a) must all be the same number.
- Calling every break a vertical asymptote. A hole is removable and a jump has finite one-sided limits, while an infinite discontinuity has values that grow without bound.
- Canceling factors and forgetting the original restriction. If (x - 2) cancels from a rational expression, x = 2 may still be excluded from the original function and can create a removable discontinuity.
Practice Questions
- 1 For f(x) = (x^2 - 9)/(x - 3), x != 3, identify the type of discontinuity at x = 3 and find lim x -> 3 f(x).
- 2 Let f(x) = 2x + 1 for x < 1 and f(x) = x^2 + 4 for x >= 1. Find lim x -> 1- f(x), lim x -> 1+ f(x), and state the type of discontinuity at x = 1.
- 3 A graph has an open circle at (2, 5), a filled dot at (2, 1), and the curve approaches y = 5 from both sides. Explain whether the function is continuous at x = 2 and name the type of discontinuity.