1.4 Continuity and One-Sided Limits

Objective: Determine continuity at a point and on an open interval; determine one-sided limits and continuity on a closed interval.

Miss BattagliaAB/BC Calculus

What does it mean to be continuous?Continuity

Below are three values of x at which the graph of f is NOT continuousAt all other points in the interval (a,b), the graph of f is uninterrupted and continuous

f(c) is not defined

does not exist

Definition of ContinuityContinuity at a Point: A function f is continuous at c if the following three conditions are met.

1. f(c) is defined

2. exists

3.

Continuity on an Open Interval: A function is continuous on an open interval (a,b) if it is continuous at each point in the interval. A function that is continuous on the entire real line (-∞,∞) is everywhere continuous.

Removable (f can be made continuous by appropriately defining f(c)) & nonremovable.

Discontinuities

Removable Discontinuity

Nonremovable Discontinuity

Continuity of a Function Discuss the continuity of each function

One-Sided Limits and Continuity on a Closed Interval• Limit from the right

• Limit from the left

• One-sided limits are useful in taking limits of functions involving radicals (Ex: if n is an even integer)

A One-Sided Limit Find the limit of as x

approaches -2 from the right.

One sided limits can be used to investigate the behavior of step functions. A common type is the greatest integer function defined by

= greatest integer n such that n < x◦ Ex: = 2 and = -3

Find the limit of the greatest integer function as x approaches 0 from the left and from the right.

The Greatest Integer Function

𝑓 (𝑥 )=⟦𝑥 ⟧

Definition of Continuity on a Closed IntervalA function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and and

The function f is continuous from the right at a and continuous from the left at b.

Theorem 1.10: The Existence of a Limit

Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L iff

and

Continuity on a Closed Interval Discuss the continuity of

Theorem 1.11 PROPERTIES OF CONTINUITYIf b is a real number and f and g are continuous at x=c, then the following functions are also continuous at c.1. Scalar multiple: bf

2. Sum or difference: f + g

3. Product: fg4. Quotient: , if g(c)≠0

By Thm 1.11, it follows that each of the following functions is continuous at every point in its domain.

THEOREM 1.12 CONTINUITY OF A COMPOSITE FUNCTIONIf g is continuous at c and f is continuous at g(c), then the composite function given by is continuous at c.

Theorem 1.13 INTERMEDIATE VALUE THEOREMIf f is continuous on the closed interval [a,b], and k is any number between f(a) and f(b), then there is at least one number in c in [a,b] such that

f(c) = k

Consider a person’s height. Suppose a girl is 5ft tall on her thirteenth bday and 5ft 7in tall on her fourteenth bday. For any height, h, between 5ft and 5ft 7in, there must have been a time, t, when her height was exactly h.

The IVT guarantees the existence of at least one number c in the closed interval [a,b]

Intermediate Value Theorem

An Application of the IVT Use the IVT to show that the polynomial function

f(x)=x3 + 2x – 1 has a zero in the interval [0,1]

AB: Page 78 #27-30, 35-51 odd, 69-75 odd, 78, 79, 83, 91, 99-102

BC: Page 78 #3-25 every other odd, 31, 33, 34, 35-51 every other odd, 61, 63, 69, 78, 91,99-103

Classwork/Homework