Recently I decided to compare continuity, related notions, and differentiability. Can a function be differentiable, but not continuous? What about uniformly continuous, but not differentiable? I used Maplesoft's new online product, Maple Learn (free to use at learn.maplesoft.com), to explore.
Here is a Maple Learn document I created. It is an organizational diagram, as shown below. Each rectangle in the diagram corresponds to a different property that a function may satisfy. Within each rectangle, examples are provided of functions satisfying the appropriate properties.
If you click on an example, it will be selected, and the corresponding function will be plotted in Maple Learn's context panel. Try it!
I've also created companion documents to explain certain concepts in greater detail. For instance, below is a snapshot of a document explaining uniform continuity, which you can access here.
By using sliders in the document, you can move and resize the rectangle drawn in the graph. You should notice when doing this that the green function never touches the horizontal sides of the rectangle. This turns out to be the "reason" why the function is uniformly continuous.
You can find a companion document on Lipschitz continuity here.
I’ve learnt a lot about continuity in creating the documents shown. I hope that you too have learnt something from them!