>>12831842The - proofs of continuity at a point are a game you play against a function . The function throws you an error bound that can be any positive real number. You want to throw a back at the function. You win the game if .
That’s a lot of notation, but what does it mean? It’s a way of saying this: I’m taking a small neighborhood about some point in the function . I win the game (aka my function is continuous) if i find a small “nudge” against that everything within the nudge is “captured” in the small neighborhood when I map the nudge via . What this means is that small changes in can only yield certain types of small changes in , so much so that if are close, then it necessarily follows that have to be close as well. If they weren’t, then there had to have been some “jump” in where the function broke, since otherwise there’d have been an unbroken road from to by taking the values between and and mapping them via f.
The topological definition of continuity is much cleaner but equivalent to epsilon-delta in the metric topology. I feel like it makes continuity even clearer.
