This is probably one of the biggest question I have when learning some mathematics. I always wonder if I have a concept in my head lets say continuity. Lets I want this concept to be able to characterize a function that has "no breaks"; it is completely connected. So lets say I then create the a definition of continuity just as the limit definition of continuity is. And then I make theorems, lemmas, etc off of that definition. So after I am done what I learned from these theorems I can now see is implied to any function I have that has no breaks.
My question is how am I to be sure whenever I make definition that it matches up with the concept I am thinking. Like in the example above if for some reason my definition didn't imply my concept of continuity I was trying to form the new knowledge I have from my proofs would be falsely applied to concepts that it wasn't actually talking.
This goes to my overall problem with how am I to be sure the mathematical definition match up the intuition/concepts I am thinking of in reality. One way I thought to reassure my self if mathematical definition matches up with reality is think of attributes of my idea as prove my definition implies these attributes. Also I am not saying that for my example that every continuous function has to be thought as one with no breaks but at the very least it must imply that.
This also becomes more critical to me when the definitions become less intuitive and more abstract