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 

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

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

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

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

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

Non-removable discontinuity Ex. -1 1

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

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

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

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

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)

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

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

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

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