Calculus Limits and Continuity

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What is a continuous function?

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A function is continuous if there are no breaks, jumps, or holes in its graph. Formally, a function f(x) is continuous at a point c if: 1) f(c) is defined, 2) the limit of f(x) as x approaches c exists, and 3) the limit equals f(c).

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What is a removable discontinuity?

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A removable discontinuity occurs when a function is not defined at a point, but can be made continuous by defining or redefining the function at that point. It typically appears as a hole in the graph.

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What is a jump discontinuity?

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A jump discontinuity occurs when the left-hand limit and the right-hand limit at a point exist but are not equal. This results in a 'jump' in the graph of the function.

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What is an infinite discontinuity?

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An infinite discontinuity occurs when the function approaches infinity (or negative infinity) as it approaches a certain point. This typically happens with vertical asymptotes.

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What is the limit of a function?

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The limit of a function describes the value that the function approaches as the input approaches a certain point. It can be evaluated from the left (left-hand limit) and from the right (right-hand limit).

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What must be true for a limit to exist?

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For a limit to exist at a point, the left-hand limit and the right-hand limit must be equal.

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What is the difference between left-hand limit and right-hand limit?

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The left-hand limit is the value that a function approaches as the input approaches a certain point from the left side, while the right-hand limit is the value that the function approaches as the input approaches from the right side.

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How do you classify a function at x = 0 if it is continuous?

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If a function is continuous at x = 0, it means that the function is defined at that point, the limit exists, and the limit equals the function's value at that point.

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What is the significance of continuity in calculus?

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Continuity is significant in calculus because it ensures that functions behave predictably, allowing for the application of various theorems and techniques, such as the Intermediate Value Theorem.

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What is the Intermediate Value Theorem?

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The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], then it takes on every value between f(a) and f(b) at least once.