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6.2. Continuous Functions
Java Applet: Illustrating Continuity
Context
This graph shows the function
 f(x) = x^{4} + 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 xaxis
 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) = x^{2} + 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 xaxis
 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.