Continuity

Recall that a function

is continuous at a point

if for every

there exists a

such that if

, then

. Further, we say that

is uniformly continuous if for every

there exists a

such that if

, then

. This visualization allows one to fix an

and adjust

so that the continuity conditional is satisfied. In addition, one can move the point

on the

-axis to see the effect on

and explore the idea of uniform continuity.

Continuity and Uniform Continuity

Using the applet work on the following questions:

Starting with position the point and let , what is the largest value of that will satisfy the continuity conditional?
What is the effect of moving to the right?
Change the function to . Position the point and let , what is the largest value of that will satisfy the continuity conditional? What is the effect of moving the point on ?
What is it about these two functions that causes the different effect on ?
Returning to the function , for a particular choice of can we ever find a that will work for all ? What about the function ?
Again, returning to , if we restrict to the interval can we find a single that will work for all ?
Consider the function . For a given can we find a single that works for all ? What about for ? What's the difference?