I don't understand uniform continuity :( I don't understand what uniform continuity means precisely. I mean by definition it seems that in uniform continuity once they give me an epsilon, I could always find a good delta that it works for any point in the interval, but I don't understand the significance of this definition and I don't know how I could study if a function is uniformly continuous on an interval or not. For some functions it's easy, for example I've already proved that the function f(x)=ax+b is uniformly continuous over R for any a,b, because I could always choose delta to be epsilon/a and it doesn't depend on the point that I'm writing down the definition of continuity for it. I think I need to see more examples. If someone writes down a lot of examples here and proves that some of them are uniformly continuous while the others aren't I'll be happy :). I need to see how the definition works in real-situation examples. I hope I'm not asking too much. Thanks in advance.