2.5 Continuity

"Intuitively" - Draw without lifting your pencil

Examples where a function is NOT continuous at a

I. II. III. IV

I. does not exist

II. f(x) is not defined at a

III. f(a) ¹

IV. f(x) is not defined at a

DEFINITION OF CONTINUITY

A function f is continuous at a number a if = f(a)

This definition has THREE requirements:

1) f(a) is defined

2) exists

3) =f(a)

Note: If 3) is true, so are 1) and 2) !!!

Types of Discontinuities

Removable discontinuity #II and III above

Jump discontinuity #I above

Infinite discontinuity #IV above

Examples: Identify where each of the function is continuous.

Specify any discontinuities as to type.

Definition:

A function f is continuous from the right of a number a if = f(a)

A function f is continuous from the left of a number a if = f(a)

Example: The step function is continuous from the right at each integer.

Definition:

A function f is continuous on an interval if it is continuous at every number in the interval.

Use the definition of continuity and the properties of limits to show that the function is continuous on the interval

#38 Find the numbers where f is discontinuous. At each of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f.

Theorem 4

If f and g are continuous at a and c is a constant, then the following functions are also continuous at a:

1. f + g 2. f - g 3. cf 4. fg 5.

Theorem 5

a) Any polynomial is continuous everywhere; that is, it is continuous on

b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.

Trig Functions

Theorem 7

The following types of functions are continuous at every number in their domains:

Polynomials; rational functions; root functions; trigonometric functions

Theorem 8

If f is continuous at b and , then

That is:

Theorem 9

If g is continuous at a and f is continuous at g(a), then the composite function f(g(x)) is continuous at a.

Example:

So f(x) = g(h(x)) is continuous only on the part of h(x) that causes g(h(x)) to be continuous, that is where 16 - x > 0 or (- ¥,16]

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

#22 #24

#26 #28

#38 Find the values of a and b that make f continuous everywhere

The Intermediate Value Theorem

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

Real live example:

A baby is 21” long at birth and 3 feet (36”) at 4 years old. Since height is a continuous function, then at some age between newborn and four, the child must have been 30” tall.

A consequence of the this theorem is that if f(a) and f(b) have opposite signs, then there is a number c such that f(c)=0; that is f has a zero on the [a, b].

Example:

Verify the intermediate value theorem for has a solution between 0 and 1.

#48 Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

Assignment; 2.5, page 105, 1, 3, 4, 5, 7, 11, 12, 13-27 odd, 31-41 odd,

43 a & b, 47, 49, 53, *61(Hint: IVT)