Hi Scholars!

This week we consider uniform continuity, specifically the following result.

Lemma: Every continuous function on a closed and bounded domain is uniformly continuous.

In my experience, the idea of uniform continuity is often confusing for undergrads. They initially cannot tell the difference between the choice of δ for a continuous and a uniformly continuous function. Of course, one must emphasize that in the case of uniform continuity δ does not depend on the point z of interest. The simplest way I found to illustrate this is with the graphic above, which is of a quadratic function on an interval [a,b], with a > 0. By animating the graphic, we can see that the same δ can be used for all points in the domain. We can also comment that, given any positive ε, there is a positive δ such that any pair of points within δ of each other will have function values within ε of each other. In other words, the function should exit the blue rectangle on the sides (green dashed lines) rather than the top or bottom (red dashed lines). Some algebra is included at the top of the diagram to show an explicit formula for δ, which is solely a function of ε and b.

What do you think? Are there other ways this concept might be illustrated well?

Stay Awesome.

Howard