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What is Continuity?

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Used in a sentence

University Course Reader · STEM

Because the function is continuous on the closed interval, the intermediate value theorem guarantees a root.

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Overview

A function is continuous if it has no jumps, holes, or breaks, so you could draw it without lifting your pen. At each point, the value the function approaches must match the value it actually has. Where those disagree, the function is discontinuous.
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Overview

Continuous means you can draw the whole thing without lifting the pen. No holes, no jumps, no teleporting from one value to another. The second a function leaps with nothing in between, it's discontinuous. 😎

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Detail

Continuity is checked one point at a time, and three things must hold there. The function has a value at the point, the graph closes in on a single number as you approach from both sides, and that number matches the function's actual value. If the graph closes in on a number but the value there is missing or different, you get a hole. If the two sides close in on different numbers, you get a jump, the way postage jumps to a higher rate at 501 grams. Most of calculus assumes continuity, and anything with a derivative must be continuous, though a continuous curve can still have sharp corners with no derivative. There is also a useful guarantee. A continuous function below some number at one end and above it at the other must hit that number in between, so a solution exists before you find it.
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Detail

Three boxes to tick at a point: the function has a value there, the curve closes in on one number from both sides, and that number matches the value it actually has. Miss the last box and you get a hole, one missing pixel in an otherwise fine curve. Miss the both-sides-agree part and you get a jump, which every pricing table on earth does on purpose. Calculus is built for smooth things, so anything with a derivative has to be continuous first, though continuous doesn't mean smooth: a sharp corner still counts, it just has no single slope at the tip. The bonus guarantee: below a number at one end, above it at the other, then somewhere in between it hits that number exactly, so you know an answer exists before you go hunting for it. 😎

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Analogy

Two crews dig a tunnel from opposite ends of a mountain, aiming to meet in the middle. The tunnel works only if both halves arrive at the exact same spot and the rock at the meeting point is actually cut through: that is continuity at a point. Halves arriving at different heights are a jump, an uncut stretch where they meet is a hole, and either way nobody drives through.
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Analogy

A zipper on your jacket. It closes only if, at every tooth, the two sides come together at the same spot and a tooth is actually sitting there. One missing tooth is a hole and it jams, two sides sewn on crooked that never line up is a jump, and either way you spend the day holding your jacket shut.
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AI explanations may contain errors · Not professional advice

Formal definition — The same term, explained the usual way

A function is continuous at a point when the limit of the function as the argument approaches that point exists and equals the function's value there. Continuity on an interval underpins the intermediate value and extreme value theorems, and differentiability at a point implies continuity, though continuity does not imply differentiability.

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