Continuity

Section 2.3

Continuity

• Application of limits• A function f is continuous at x = a if

o f(x) existso The limit of f(x) existso

)()(lim afxfax

Continuity at a Point

• Continuous functions have graphs that can be sketched in one continuous motion without lifting your pencil.

• The outputs vary continuously with the inputs and don’t jump from one value to another without taking on the values in between.

See p. 74

Interior Point:A function f is called continuous at x = c if c is in its domain and

limx c

f (x) f (c)

Endpoint:A function y= f(x) is continuous at a left endpoint a or is continuous at a right endpoint b of its domain if

limx a

f (x) f (a) or limx b

f (x) f (b)

If a function is not continuous at a point c, then f is discontinuous

at c, and c is a point of discontinuity of f.

(c doesn’t have to be in the domain of f)

Types of Discontinuities

• Removable• Jump• Infinite• Oscillating

See p. 76

Page 77 Exploration 1

Continuous Functions

• A function is continuous on an interval if and only if it’s continuous at every point on the interval.

• A continuous function is one that’s continuous at every point of its domain. (It doesn’t have to be continuous on every interval.)

• Page 78 Properties of Continuous Functions• Composites of continuous functions are

continuous.

Intermediate Value Theorem for Continuous Functions

• A function y = f(x) that is continuous on a closed interval [a, b] takes on every value between f(a) and f(b). In other words, if d is between f(a) and f(b), then d = f(c) for some c in [a, b] .

pages 80-81 (2-30 even, 44)

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pages 80-81 (3-9 odd, 19-29 odd)