Continuity

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What is continuity in mathematics?

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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 at a point if the limit as it approaches that point equals the function's value at that point.

Tags

CCSS.HSF-IF.C.7D

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What does it mean for a function to be continuous at a point x = a?

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

Tags

CCSS.HSF.IF.A.2

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What is removable discontinuity?

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Removable discontinuity occurs at a point where a function is not defined or has a hole, but can be made continuous by defining or redefining the function at that point.

Tags

CCSS.HSF-IF.C.7D

Front

What is jump discontinuity?

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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 function's value.

Tags

CCSS.HSF-IF.C.7D

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What is infinite discontinuity?

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Infinite discontinuity occurs when the function approaches infinity or negative infinity as it approaches a certain point, resulting in a vertical asymptote.

Tags

CCSS.HSF-IF.C.7D

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What is the definition of a continuous function over an interval?

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A function is continuous over an interval if it is continuous at every point within that interval.

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Provide an example of a function that is continuous everywhere.

Back

The function f(x) = x^2 is continuous everywhere on the real number line.

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Provide an example of a function with a removable discontinuity.

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The function f(x) = (x^2 - 1)/(x - 1) has a removable discontinuity at x = 1, where it can be redefined to be continuous.

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What is the limit of a function?

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The limit of a function at a point is the value that the function approaches as the input approaches that point.

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How do you determine if a function is discontinuous at a point?

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A function is discontinuous at a point if any of the following conditions are met: 1) the function is not defined at that point, 2) the limit does not exist, or 3) the limit does not equal the function's value at that point.