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?