6.2. Continuous Functions
Java Applet: Illustrating Continuity
Context
This graph shows the function
- f(x) = x4 + 0.5
if x < 1
- f(x) = 0.5 x + 1
if 1
 x < 2
- f(x) = cos(
 x) + 1
if 2 x
|
To use the Java applet:
- Click on Options for control panel
- Select x by clicking on the x-axis
- Select a given epsilon
- Guess delta and press Apply
|
If for
any given epsilon value you can find a positive delta
value such that
the red area touches the graph inside the green area,
then your function is continuous at the given values of
x
This graph shows the function
- f(x) = x2 + 1.3
if x < 1
- f(x) = sin(x - 1) + 2
if 1 x
|
To use the Java applet:
- Click on Options for control panel
- Select x by clicking on the x-axis
- Select a given epsilon
- Guess delta and press Apply
|
If for
any given epsilon value you can find a positive delta
value such that
the red area touches the graph inside the green area,
then your function is continuous for
x = c
The first function is continuous for all x, while the second
function is not continuous for x = 1 and continuous for all
other x.
Compare with uniform continuity.
Interactive Real Analysis