Continuity

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What is the definition of continuity at a point?

Back

A function f is continuous at a point c if: 1) f(c) is defined, 2) \( \lim_{x \to c} f(x) \) exists, and 3) \( \lim_{x \to c} f(x) = f(c) \).

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What is the Intermediate Value Theorem?

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If f is continuous on the interval [a, b] and N is any number between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = N.

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What does it mean for a function to be uniformly continuous?

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A function f is uniformly continuous on an interval if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for all x, y in the interval, if \( |x - y| < \delta \), then \( |f(x) - f(y)| < \epsilon \).

Front

State the definition of a limit.

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The limit of f(x) as x approaches c is L if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - c| < \delta \), then \( |f(x) - L| < \epsilon \).

Front

What is the relationship between continuity and limits?

Back

A function f is continuous at a point c if \( \lim_{x \to c} f(x) = f(c) \). If a function is not continuous at c, then the limit does not equal the function value.

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What is the definition of a compact set in the context of continuity?

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A set is compact if it is closed and bounded. The continuous image of a compact set is also compact.

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What is the Extreme Value Theorem?

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If f is continuous on a closed interval [a, b], then f attains a maximum and a minimum value on that interval.

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What does it mean for a function to be continuous on an interval?

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A function f is continuous on an interval if it is continuous at every point in that interval.

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What is the definition of an open set in topology?

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A set U in a metric space is open if for every point x in U, there exists a radius r > 0 such that the ball of radius r centered at x is entirely contained in U.

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What is the definition of a closed set in topology?

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A set C is closed if it contains all its limit points, or equivalently, if its complement is open.