ContinuityChapter 2: Limits and Continuity

What you’ll learn about

• Continuity at a Point• Continuous Functions• Algebraic Combinations• Composites• Intermediate Value Theorem for Continuous

Functions

Continuity at a Point Any function whose graph can be sketched in one continuous motion

without lifting the pencil is an example of a continuous function.

y f x

Interior Point: A function is continuous at an interior point of its

domain if lim

Endpoint: A function is continuous at a left

endpoint or is continuous

x c

y f x c

f x f c

y f x

a

at a right endpoint of its domain if

lim or lim respectively.x a x b

b

f x f a f x f b

This function to the left is continuous at a, b and c. At c, there is a two-sided limit that is equal to f(c) and at both a, and b, there are the appropriate one-sided limits that are equal to f(a) and f(b), respectively. Since a and b are endpoints all we need is the one-sided limit to prove continuity.

Continuity at a PointIf a function f is not continuous at a point c , we say that f is discontinuous at c and c is a point of discontinuity of f.

Note that c need not be in the domain of f.

There are four major types of discontinuities at a point that we will deal with in the class. They are jump, removable, infinite, and oscillating.

Example 1 - Continuity at a Point

2

3Find and identify the points of discontinuity of

1y

x

There is an infinite discontinuity at 1.x

Now let’s look at examples of all four main types of discontinuities on the next slide.

Continuity at a Point

• a – no discontinuity

• b– removable discontinuity

• c – removable discontinuity

• d – jump discontinuity

• e – infinite discontinuity

• f – oscillating discontinuity

*Functions are considered continuous functions even though they are not continuous over the whole number line. For example, is considered continuous. Note, however, that is only continuous for because that is its domain. In general, functions are considered continuous if they are continuous over their domain.

* is NOT continuous over the interval [-1,1] even though it is a continuous function in general. This is extremely confusing for a lot of students. For the sake of this class, we are almost exclusively concerned with continuity over a given interval so just focus on those specific intervals when answering problems.

Continuous FunctionsA function is continuous on an interval if and only if it is continuous at every point of the interval. A continuous function is one that is continuous at every point of its domain. A continuous function need not be continuous on every interval.

Example 2 - Continuous Functions

The given function is a continuous function because it is

continuous at every point of its domain. It does have a

point of discontinuity at 2 because it is not defined there.x

2

2

2y

x

Example 3 – Removing a DiscontinuityRemovable discontinuities are called that because you can actually remove them from the function by creating an extension of your original function. Think of it as just plugging the tiny whole at the spot where you have a removable discontinuity.

Let’s find any remove any discontinuities that we can from the following function by creating a continuous extension of the original function.

Properties of Continuous Functions

If the functions and are continuous at , then the

following combinations are continuous at .

1. Sums :

2. Differences:

3. Products:

4. Constant multiples: , for any number

5. Quotients: , pr

f g x c

x c

f g

f g

f g

k f k

f

g

ovided 0g c

Just like limits at a point and limits at infinity, continuous functions have set of basic rules for combining functions.

Composite of Continuous Functions If is continuous at and is continuous at , then the

composite is continuous at .

f c g f c

g f c

*See Example 4 on page 82 of your book. It uses most of the rules from this slide and the last slide.

Intermediate Value Theorem for Continuous Functions

0 0

A function that is continuous on a closed interval [ , ]

takes on every value between and . In other words,

if is between and , then for some in [ , ].

y f x a b

f a f b

y f a f b y f c c a b

f(b)

y0

f(a)

Intermediate Value Theorem for Continuous Functions

The Intermediate Value Theorem (I.V.T. for short) for Continuous Functions is the reason why the graph of a function continuous on an interval cannot have any breaks. The graph will be connected, a single, unbroken curve. It will not have jumps or separate branches.

The IVT is an existence theorem. That means all it can do is guarantee that something exists within a certain interval. It can’t tell is the exact x value where it actually occurs. It is often used to prove that a root or zero exists within a given interval.

Example 4 – I.V.T.Does have a zero in the interval [3, 4]?

and We know that f(x) changes sign on the interval [3, 4].We also know that f(x) is continuous because it is a polynomial. Since f(x) is a continuous function and changes sign on the interval [3, 4], by the IVT there must be a 0 in that interval.

Summary• Definition: is continuous at if • Right-continuous at if • Left-continuous at if

• If is continuous at all points in its domain, f is simply called continuous.• There are four common types of discontinuities: removable, jump, infinite, and

oscillating.• A removable discontinuity can often be fixed using an extension of the original function.• There are properties of continuity: sums, products, multiples, differences, quotients

(when the denominators ≠ 0) and composites are also continuous.• Basic functions: Polynomials, rational functions, nth-root and algebraic functions, trig

functions and their inverses, exponential and log functions are continuous on their domains.

• The Intermediate Value Theorem can be used to determine if a certain f(x) value must exist over a certain interval.