Even though not all functions are continuous on all real numbers (i.e., everywhere), we can still talk about the continuity of a function in terms of intervals. As you may recall, there are three kinds of intervals: open intervals, closed intervals, half-open (or half-closed) intervals.

Definition of f(x) is continuous on an open interval:We say a function f(x) is continuous on an open interval of (a, b) if there are no points of discontinuity on the interval (a, b). That is, f(x) is continuous at any x where a < x < b.

We say a function f(x) is continuous on a closed interval of [a, b] if i. f(x) is continuous on the open interval (a, b), ii. f(a) and f(b) are defined, and iii. limxa f(x) = f(a) and limxb f(x) = f(b).

We say a function f(x) is continuous on a half-open interval of [a, b) if i. f(x) is continuous on the open interval (a, b), andii. f(a) is defined and limxa f(x) = f(a).

We say a function f(x) is continuous on a half-open interval of (a, b] if i. f(x) is continuous on the open interval (a, b), andii. ______________________________.

Page 1Intervals of Continuity

+ –

+

ba a b ba ba

ba

ba ba

Page 2Continuous on its Domain

Recall that functions such as f(x) = x1/2 and f(x) = log x are not continuous on all real numbers, nevertheless, they are continuous at every number in their domains. For example, the domain of f(x) = x1/2 is [0, ) and its interval of continuity is also [0, ). Similarly, the domain of f(x) = log x is (0, ) and its interval of continuity is also (0, ). Another example is f(x) = 1/x, where its graph has two pieces and there is a vertical asymptote at x = 0. However, since its domain is all real numbers except 0, i.e., (, 0) (0, ), and we can see that the left piece of f(x) = 1/x is continuous on (, 0) where as the right piece is continuous on (0, ), we still say that the function is continuous on its domain.

So what is an example of a function which is not continuous on its domain?There are many functions which are not continuous on their domains. For example, the integer function, f(x) = [x]. The domain is all real numbers (plug any real number x into the integer function, it will yield back the greatest integer ≤ x). However, the function is not continuous on its domain since it is discontinuous at every integer. Another example is the sign function,

f(x) = sgn x = where given any real number x, this function will

yield back 1 if x is positive, 1 if x is negative, and 0 is x is zero. As you can see, its domain is all real numbers, but it’s not continuous at x = 0.

1, 0

0, 0

1, 0

x

x

x

Page 3How to Give the Intervals of Continuity of a Function

If a function is continuous on all real numbers, e.g., f(x) = x2, then we say its interval of continuity (IOC) is (, ).

If a function is continuous on its domain, then its interval(s) of continuity is same as its domain. For example, i) f(x) = x1/2 IOC = [0, ); ii) f(x) = 1/x IOC = (, 0) (0, ).

If a function is not continuous on its domain, e.g., f(x) = sgn x, we must say the IOC is (, 0) (0, ) despite that the domain is (, ). This implies the interval notation for the domain is not necessarily same as the intervals of continuity.

ExampleFor the function with graph below, give the domain and intervals of continuity using the least number of intervals as possible.

Domain: ____________________________IOC: _______________________________

3

2

1

0–1

–2

–3

1 2 3 4 5 6 7 8 9 10 11–1

–2–3

–4

–5

–6

y = f(x)

Domain: (-oo, -1) U (-1,5) U (5, oo)IOC: (-oo, -4] U (-4,-1) U (-1, 1)U (1, 3) U (3, 5) U (5, 8) U [8.oo)

Page 4Properties of Functions which are Continuous on their Domains

ExampleLet f(x) = sin x and g(x) = x2 – 1 . Find the intervals of continuity of the following functions:1. f(x) + g(x)2. f(x) – g(x) 3. f(x)∙g(x) 4. f(g(x))5. g(f(x))6. f(x)/g(x)7. g(x)/f(x)

Theorem:The following types of functions are continuous at every number in their domains:i) polynomial functions ii) rational functions iii) root functionsiv) trigonometric functions v) exponential functions vi) logarithmic functionsvii) absolute-value functions viii) inverse trigonometric functions

Properties:If f(x) and g(x) are two functions continuous at every number in their respective domains, and let D be the domain of f(x) + g(x) and E be the domain of f(g(x)), then:i)f(x) + g(x), f(x) – g(x) and f(x)∙g(x) are continuous on D,ii)f(x)/g(x) is continuous on D – {x | g(x) = 0}, and iii)f(g(x)) is continuous on E.

Continuous on all real numbers

Continuous only on their domains

Polynomial Rational

Odd-indexed root Even-indexed root

sin, cos tan, cot, sec, csc

tan–1, cot–1 sin–1, cos–1, sec–1, csc–1

Exponential Logarithmic

Absolute-value

Page 5

There are several ways we can find the limit of a function f as x approaches a without knowing its graph. One obvious way is to use numbers close to a. See the following examples:

How Do We Find the Limit of a Function Without the Graph?

x 0.9 0.99 0.999 1 1.001 1.01 1.1

x2 + 1

1. limx1 (x2 + 1) = ___x 1.9 1.99 1.999 2 2.001 2.01 2.1

1/x2

2. limx2 1/x2 = ___

x 2.9 2.99 2.999 3 3.001 3.01 3.1

3. limx3 (x – 3)/|x – 3| = ___ 4. limx2 1/(x2 – 4) = ___

3

| 3 |

x

x

x 1.9 1.99 1.999 2 2.001 2.01 2.1

2

1

4x

x –1.1 –1.01 –1.001 –1 –.999 –.99 –.9

5. limx–1 5/(x + 1) = ___ 6. limx0 6 sin x/x = ___ (where x in radians)x –.1 –.01 –.001 0 0.001 0.01 0.1

5

1x 6sin x

x

x 1.9 1.99 1.999 2

7. limx2– 7/(x – 2) = ___

7

2x

8. limx2+ 8/(x + 2) = ___

x 2 2.001 2.01 2.1

8

2x

9. limx∞ 3x/(2x – 9) = ___

x 100 1,000 10,000 ∞

3

2 9

x

x

Page 6Implication of Continuity on Limits

Recall that if f(x) is continuous at x = a, then the limit of f(x) as x approach a must exist and it must equal to f(a). Therefore, when we need to evaluate limxa f(x) and if we know f(x) is continuous at x = a, all we need to do is to evaluate f(a), i.e., whatever f(a) is, is the limit!

Direct Substitution Property:If f(x) is continuous at x = a, then limxa f(x) = f(a).

With this property, it allows us to evaluate the limit by using the so-called “plug-in” method. That is, as long as we know f(x) is continuous at a, even if we don’t know the graph of f(x), we can just evaluate f(a) and that will be limit! We don’t need to use any numbers close to a, hence, it is a much better and faster way of finding the limit than the “tabular” method used on page 5.

Examples:

1. limx1 (x2 + 1) = 2

2. limx2 1/x2 = 1/22 = ¼

3. limx3+ (x2 – 2)/(x + 1) = 7/4

4. limx– cos x =

Rule of Thumb of Evaluating Limit When you evaluate the limit of a function f(x) as x approaches a, just plug in a into the f(x) first. That is, when limxa f(x) is asked, just do f(a).

The rule of thumb above really is another way (actually, my way) to state the Direct Substitution Property. The difference is: here we don’t even need to care whether f(x) is continuous at a or not.

By Direction Substitution Property By My Rule of Thumb

Example 1: x2 + 1 is a quadratic function. Quadratic functions are a type of polynomial functions and polynomial functions are continuous everywhere, therefore it’s okay to plug the 1 into x, and obtain 12 + 1 = 2 as the limit.

Example 1: 12 + 1 = 2

Example 2: 1/x2 is a rational function with a vertical asymptote at x = 0. So 1/x2 is discontinuous at x = 0 but it’s continuous elsewhere, therefore it’s okay to plug the 2 into x, and obtain 1/22 = ¼ as the limit.

Example 2: 1/22 = 4