Continuity

Definition

is continuous at

This also implicitly requires that:

1. is defined
2. exists

A function is continuous on an interval it is continuous at every number in that interval. Continuity has sidedness. A function can be continuous from the left or right if the limit is one sided.

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Laws of Continuity

1. If and are continuous at then the following are continuous at .
2. is continuous at and is continuous at is continuous at
3. is continuous at and

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Continuous Functions

Polynomials, Rational, Root and Trigonometric Functions are all continuous for all numbers in their domain.

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Intermediate Value Theorem

is continuous on and

Proof: as if we can consider the function then so and if the theorem holds for meaning then and . Define and . We claim that . Note that is not empty as and since we get that . Assume for contradiction that . Then we can split into 2 cases.

Case 1 First note that by definition this means that . Since is a lower upper bound we have that . But as and we have that so . Then by continuity at , . Consider the case with , we can pick a suitable with so by continuity we must have a contradiction.

Case 2 We can’t have as by our we have said that but . So so let . There exist infinitely many so we can make arbitrarily small. By continuity at , . Consider , pick so that then we must have a contradiction of .

The only valid case left is .

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Extreme Value Theorem

is continuous on attains a global maximum and minimum value on