Continuity

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What is continuity in mathematics?

Back

Continuity refers to a property of a function where it is uninterrupted or unbroken at a point or over an interval. A function is continuous if small changes in the input result in small changes in the output.

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What are the types of discontinuities?

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1. Removable Discontinuity: A point where a function is not defined, but can be made continuous by defining it appropriately. 2. Jump Discontinuity: A point where the left-hand limit and right-hand limit exist but are not equal. 3. Infinite Discontinuity: A point where the function approaches infinity.

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What is a removable point discontinuity?

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A removable point discontinuity occurs when a function is not defined at a certain point, but the limit exists at that point. It can be 'removed' by redefining the function at that point.

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What is an infinite discontinuity?

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

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What is a jump discontinuity?

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

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How can you determine if a function is continuous at a point?

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A function f(x) is continuous at a point x = c if: 1. f(c) is defined. 2. The limit of f(x) as x approaches c exists. 3. The limit equals f(c).

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What does the notation (-∞, -2) ∪ (-2, ∞) represent?

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This notation represents the intervals where a function is continuous, indicating that the function is continuous for all real numbers except at x = -2.

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Is the function f(x) = x² - 2x + 1 continuous on the interval [-7, 7]?

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TRUE. This polynomial function is continuous everywhere, including the interval [-7, 7].

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What does the notation [-2, 2] indicate?

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This notation indicates a closed interval from -2 to 2, meaning the function is continuous at all points between -2 and 2, including the endpoints.

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What is the significance of limits in determining continuity?

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Limits help determine continuity by showing the behavior of a function as it approaches a certain point. If the limit exists and equals the function's value at that point, the function is continuous.