I am not asking for any particular question today rather I need some help to understand the concept of a function that is not uniformly continuous. Ok, so If I understand this correct, to prove a function that is not uniformly continuous, we have to find some $\epsilon>0$ and some $x,y$ within their domain such that these violate the definition of uniform continuity(if I want to get a contradiction), right? But, my problem is, I am always getting stuck when finding such $x$ and $y$. For example, you can see I asked this question here yesterday, and my choices of $x$ and $y$ weren't the correct ones, and the idea I've got from the answers and the comments is that choosing $x$ or $y$ values with $\delta$ in it might do the trick. But, that's not always the case because if you watch this youtube video of a function that is not uniformly continuous, you can see the instructor didn't choose the $y$ value with $\delta$ in it. So, my question is, is there any rule of thumb when working with functions that are not uniformly continuous? Also, if I choose such $x$ and $y$ values with $\delta$, does it mean I am guaranteed to get a answer that can give me a contradiction?
I am new to real analysis, so I need all the help I can get to understand this subject. Also, I don't know if this question fits here, but then again I don't know where to ask such questions. Thanks so much.