Section 1.4 Continuity and One-sided Limits. Continuity – a function, f(x), is continuous at x = c only if the following 3 conditions are met: 1. is defined.

Section 1.4

Continuity and One-sided Limits

Continuity – a function, f(x), is continuous at x = c only if the following 3 conditions are met:

1. is defined

2. exists

3.

Continuous on an open interval (a, b)A function is continuous on (a, b) only if it is continuous at every point in (a, b).

f c

limx cf x

limx cf x f c

Condition 1 is not met: hole in graph

c

f(c) is not defined

Condition 2 is not met: jump or asymptote

c

does not exist

c

does not exist limx cf x

lim

x cf x

Condition 3 is not met: hole in graph and function defined elsewhere.

c

L

f(c)

limx c

f c f x

Two Types of discontinuities

1. removable – function that can be made continuous at a point by redefining f(c).

ex: hole in graph

2. nonremovable – cannot redefine f(c) to make the function continuous.

ex: asymptote or jump in graph

Examples: Discuss the continuity of each.

1.

2

1f x

x

nonrem. discont. @ x = 0 (asymptote)

Examples: Discuss the continuity of each.

2.

2 2 1

1

x xf x

x

rem. disc. @ x = 1 (redefine f(1)=0)

1

Examples: Discuss the continuity of each.

3.

3

4, 2

, 2

x if xf x

x if x

nonrem. disc. @ x = 2 (jump)

2

6

8

Examples: Discuss the continuity of each.

4.

cosf x x

cont. on (-∞, ∞)

One-sided Limits

limit from the right

limit from the left

limx c

f x L

limx c

f x L

Examples: Evaluate each limit.

5. 2

2lim 4x

x

= 0

-2

Examples: Evaluate each limit.

6. 2

2lim 4x

x

= DNE

-2

Examples: Evaluate each limit.

7. 1

limx

x = 0

1

Examples: Evaluate each limit.

8. 2

limx

x = -2

-2

Existence of a LimitWhen , then lim lim

x c x cf x f x L

lim

x cf x L

Continuity on a closed interval [a, b]

f(X) is continuous on (a, b) and

lim limx a x b

f x f a and f x f b