Dini derivative

If you're doing maths at that level, I assume you know what a limit is. As you may know, a limit is not guaranteed to exist. For instance, the limit of Sin(1/x) does not exist as x -> 0 from above. If you graph the function, you can see that as you get closer to 0, the frequency of oscillation grows too fast for a limit to exist.

In the same way, the derivative of a function is not always guaranteed to exist (since the derivative is a limit by definition). You can get around this problem by using liminf and limsup.

The supremum of a set is the least upper bound on a set (it sometimes helps to think of it as the greatest element of a set, though this is not strictly correct).
The limsup is the supremum of the set as it goes to some limit.

Hopefully this article will help

Yeah, that's pretty much my understanding of it. I'm just working from the wikipedia definition, though.
I never came across the dini derivative in college. It looks like it might be useful for analysing brownian motion. What context did you learn about it in?