# Continuity

- Author:
- Albert Schueller

- Topic:
- Continuity

Recall that a function is if for every there exists a such that if , then .
Further, we say that is 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.

*continuous*at a point*uniformly*continuous if for every## 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?