An interesting mixture between continuous and discrete math - help needed!

All,

The following definition:

involves continuous mathematics (Calculus) with discrete mathematics (the sign function, whose values are discrete).

This definition along with other interesting definitions, where continuous and discrete math join together, are depicted lucidly in the following video:

http://www.youtube.com/watch?v=mM5JY-Q6hiM

Any help with reviewing this video and the suggested definitions will be greatly appreciated!

Thanks a lot,

-amir (Happy)