Dini continuity

Okay, imagine you have a line of dominoes. If you push the first one, it will knock over the second one, which will knock over the third one, and so on.

Dini continuity is basically like a domino effect for numbers, but it has to do with how close together they are. Let's say you have a line of numbers, and they're getting smaller and smaller as you go along. If they keep getting smaller and smaller, but they never get to zero, then they are said to be "Dini continuous".

This is kind of like if you had a bunch of tiny dominoes that were all touching each other, so if you pushed the first one, they would all fall down together. The numbers are so close together that they act like one big continuous line.

When mathematicians talk about Dini continuity, they're usually looking at functions, which are like machines that turn one number into another number. If a function is Dini continuous, it means that no matter how close together two numbers are, the output of the function will also be very close together.

Overall, Dini continuity is a way of saying that a bunch of numbers, or a function that uses numbers, are very close together and act like one smooth, continuous line.