CONTINUITY – Part 2

Theorem: (7, pg. 102) The following types of functions are continuous at every number in their domains.

Polynomials Rational Functions

Root Functions Trigonometric Functions

Examples: Determine the intervals for which each function is continuous?

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Theorem (8, pg. 103) The Limit of Composite Functions:

If f is continuous at b and , then . In other words, .

Continuity – part 2, page 2

Theorem (9, pg 103) Continuity of Composite Functions:

If g is continuous at a and f is continuous at g(a) (g(a) = b), then the composite function given by is continuous at a.

Examples:

Suppose and , , and .

Note that both f and g are continuous on , and that h is continuous on then, . Determine continuity of the composite functions:

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Continuity – part 2, page 3

The Intermediate Value Theorem (IVT) (10, pg. 104)

Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where f(a) f(b). Then there exists a number c in (a,b) such that f(c) = N.

Note: Most often this is used to show that if f is a continuous function on a closed interval [a,b] and N = 0, such that f(a) is some number > 0 and f(b) is some number < 0 (or visa-versa), then there exists a number c between a and b such that f(c) = 0, or x = c is a root of the equation y = f(x).

Examples:

21. Use the IVT to show that there is a root of on the interval .

22. Prove that the equation has at least one real root.

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(Note: this is the same as the equation , Also note that no interval was given, therefore it is up to you to choose one that works.)