Continuity Continuity

Introduction

A continuous function is one we can draw without lifting our pencil. Thus:

1. The function must be defined at every point.
2. The limit of the function must be defined at every point.
3.

f(x)=\lim_{x \to a}, \forall x
.

Here are some examples of continuous functions:
1.
f(x)=x^2

2.
f(x)=\sin(x)

3.
f(x)=e^x

Functions are not continuous when they have one or more of the following:
1. A hole
(The limit exists but the function does not.
or
The function does not equal the limit.)
2. A jump
(The function exists but the limit does not.)
3. An asymptote
(The function approaches
-\infty
,
\infty
, or both.)

Examples

A hole

f(x)=\frac{\sin(x)}{x}

x=0
A jump
f(x)=\frac{|x|}{\sin(x)}

x=0
Vertical Asymptotes:
f(x)=\tan(x)

x=-\pi/2, x=\pi/2
Note that a function does not need to exist for every real number. When we define continuity, we do so for the domain of the function (Its input set).

For example, the function
f(x)=\sqrt{x}
is continuous for all
x \geq 0
.
Since the function exists only for
x \geq 0
, we say that the function is continuous.