Continuity Section 2.3

Continuity Application of limits A function f is continuous at x = a if f(x) exists The limit of f(x) exists

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

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

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