Section 1.4 Continuity The function graphed below is ...

 

Section 1.4
Continuity
A function is a continuous at a point if its graph has no gaps, holes, breaks or jumps at that
point. If a function is not continuous at a point, then we say it is discontinuous at that point.
The function graphed below is continuous everywhere.

The function graphed below is NOT continuous everywhere, it is discontinuous at x  2 and
at x  1  

We will discuss types of discontinuity later in this section. 1

Section 1.4 – Continuity

 

 

Continuity

The following types of functions are continuous over their domain.

 Polynomials, Rational Functions, Root Functions, Trigonometric Functions, Inverse
Trigonometric Functions, Exponential Functions, Logarithmic Functions

Theorem: If f and g are continuous at c, a is a real number then each are also continuous at c.
i. f + g
ii. f – g

iii. af
iv. fg
v. f/g, provided g(c)  0

Theorem: If g is continuous at c and f is continuous at g(c), then f  g is continuous at c.

Example 1: Discuss continuity for: Discontinuous: Continuous:
a. g(t)  5t5  t

b. f (x)  x2 Discontinuous: Continuous:
x2  x  6

c. h(x)  1 2x x Discontinuous: Continuous:
 cos

Section 1.4 – Continuity 2

 

 

Types of Discontinuity
Removable Discontinuity occurs when:

 lim f (x)  f(c) (or the limit exists, but f(c) is undefined.
xc

Jump Discontinuity occurs when:

 lim f (x) and lim f (x) exists, but are not equal.
xc xc

Infinite Discontinuity occurs when:

 lim f (x)   on at least one side of c. Infinite discontinuities are generally associated
xc
with having a vertical asymptote.

State the type of discontinuity at each point of discontinuity:

x = -3 x=0 x=1

And so the function is continuous over everywhere else.

One-Sided Continuity

A function f is called continuous from the left at c if lim f (x)  f (c) and continuous from
xc

the right at c if lim f (x)  f (c) .
xc

In the example above, we have continuity from the right at x = 0 and continuity on the left at 3
x = 2.

Section 1.4 – Continuity

 

 

Continuity Stated a Bit More Formally

A function f is said to be continuous at the point x = c if the following three conditions are met:

1. f(c) is defined 2. lim f (x) exists 3. lim f (x) = f(c)
xc xc

To check if a function is continuous at a point, we’ll use the three steps above. This process is
called the three step method.

Example 2: Use the 3 step method to determine continuity or discontinuity.

f (x)   x  36, x0
 x0
 x 2  6,

We need to check:

1. Is f(0) defined?

2. Check to see if lim f (x) exist. and lim f (x)
x0
x0
Must check: lim f (x)
x0 

Does lim f (x) exist? 4
x0

3. Does lim f (x) = f(0)? i.e. Compare #1 and #2 above.
x0

If at least one of the three steps fails, identify the type of discontinuity.
Section 1.4 – Continuity

 

 

Example 3: Let f ( x)  x  1, x  2 is the function continuous at x = 2?
4  x, x  2

We need to check:

1. Is f(2) defined?

2. Check to see if lim f (x) exist. and lim f (x)
x2
x2
Must check: lim f (x)
x2 

Does lim f (x) exist?
x2

3. lim f (x) = f(2)? i.e. Compare #1 and #2 above.
x2

If at least one of the three steps fails, identify the type of discontinuity.

Section 1.4 – Continuity 5

 

 

2x2  9, x  3
Example 4: Is f (x)  2,
x  3 continuous at x = 3?

 x 3 , 3 x


We need to check:

1. Is f(3) defined?

2. Check to see if lim f (x) exist.
x3

Must check: lim f (x) and lim f (x)

x3  x3

Does lim f (x) exist?
x3

3. Does lim f (x) = f(3)? i.e. Compare #1 and #2 above.
x3

If at least one of the three steps fails, identify the type of discontinuity.

Section 1.4 – Continuity 6

 

 

Try this one: Determine if the function f (x)   x  22  4 is continuous at x = 2.

We need to check:
1. Is f(2) defined?

2. Does lim f (x) exist? Must check lim f (x) and lim f (x) .
x2 x2  x2

3. Does lim f (x) = f(2)? i.e. Compare #1 and #2 above.
x2

Example 5: Find c so that f(x) is continuous. f ( x)  2x  3, x2
 x2 x2
cx  ,

1. Find f(2).

2. lim f (x) must exist, so make sure that lim f (x) = lim f (x) .
x2 x2  x2

3. Finally set lim f (x) = f(2) to find c. 7
x2

Section 1.4 – Continuity

 

 

Try this one: Find A and B so that f(x) is continuous.

2x2 1, x  2

f ( x)   A, x  2

Bx  3, x  2

1. Find f(-2).

2. lim f (x) must exist, so make sure that lim f (x) = lim f (x) .
x 2 x 2  x 2

3. Since lim f (x) = f(-2) then
x 2

Example 6: The function f (x)  x  25 is defined everywhere except at x = 25. If possible,
x 5

define f (x) at 25 so that it becomes continuous at 25.

Section 1.4 – Continuity 8

 

 

Try these:

The function f (x)  1 is defined everywhere except at x = 3. If possible, define f (x) at 3
x3

so that it becomes continuous at 3.

 1 , x2
 x
 2 x4
 4  x  6 . Find points where the function is discontinuous and classify these
 x6
 1 , x6

x


Let f (x)   x,



4,


x,




points.

Section 1.4 – Continuity 9