I have an intuitive understanding of continuity, however, I’m struggling to understand the rigorous definition.
I understand that a function is continuous when you can get infinitely close to each point of it, until you arrive at it. Basically, you can draw the whole thing without lifting your pencil.
I’m struggling to see how, however, this has been reflected in the rigorous definition lim x-> c f(x) = f(c). It seems to me that the rigorous definition is just saying that as x gets closer to c, the limit of f(x) will be f(c). Or rather, as we sub in values of x that are infinitely close to c, the value of the function becomes infinitely close to the value of f(c). But this definition is true even when the functions aren’t continuous! For instance, this definition is true for a function with a point discontinuity.
So can someone explain to me, what the formal definition means, how it is only true for continuous functions, and how it is compatible with the 2 intuitive definitions I wrote above? Thank you.
Can you also not make the explanation too rigorous? I’m just learning Khan Academy Calculus, and still haven’t touched on things like epsilon delta proofs yet.