1 6 Continuity Objectives To determine continuity of

1. 6 Continuity Objectives: To determine continuity of functions To use 3 -step definition in proving continuity of functions

What does it mean – “Continuous function”? § A function without breaks or jumps § A function whose graph can be drawn without lifting the pencil 2

To be continuous on an interval, a function must be continuous at its Every Point A function can be discontinuous at a point 1) A hole in the function and the function not defined at that point 2) A hole in the function, but the function is defined at that point 3

Continuity at a Point A function can be discontinuous at a point 3) The function jumps to a different value at a point 4) The function goes to infinity at one or both sides of the point 4

Definition of Continuity at a Point A function is continuous at a point x = c if the following three conditions are met x=c 5

Some Discontinuities are “Removable”! A discontinuity at c is called removable if … the function can be made continuous by • defining the function at x = c • or … redefining the function at x = c 6

“Removable” example Defining the function at x = 1, y=2 The open circle can be filled in to make it continuous 7

Non-removable discontinuity Ex. -1 1 8

Determine whether the following functions are continuous on the given interval. yes, it is continuous ( ) 1 9

( ) discontinuous at x = 1 removable discontinuity since filling in (1, 2) would make it continuous. Define: 10

Which of these are (Dis)Continuous when x = 1 ? … Why yes or not? Are any removable? 11

Discuss / show continuity of g(x) at x = 2 3 3 g(x) is continuous at x = 2 12

Continuity Theorem A function will be continuous at any number x = c for which is defined, when § § is a polynomial function (at every real number) is a power function (at every number in its domain) is a rational function (at every number in its domain) is a trigonometric function (in domain) 13

Properties of Continuous Functions If f and g are functions, continuous at x = c, then … § § § is continuous (where b is a constant) is continuous 14

One Sided Continuity § A function is continuous from the right at a point x = a if and only if § A function is continuous from the left at a point x = b if and only if a b 15

Continuity on an Interval (Summary) § The function f is said to be continuous on an open interval (a, b) if It is continuous at each number/point of the interval § It is said to be continuous on a closed interval [a, b] if It is continuous at each number/point of the interval, and it is continuous from the right at a and continuous from the left at b 16

Continuity on an Interval (Examples) On what intervals are the following functions continuous? 17