@user21820 yo! are you here?

damn, so hard to express myself in English, anyway what I wanted to say is that today I discovered that notion of continuity is superfluous to notion of limit lol

@famesyasd Yes if that point is in the domain of the function.

Yes!

but that is the only difference basically

$f(x)=1$ if $x=0$ and $f(x)=0$ otherwise has a limit in $0$

here I meant that we take limit including x0 in a set in which we take limits

@famesyasd: Well I'd have to say your statements are ambiguous, which is why @AlessandroCodenotti is not very satisfied haha..

I would state it as "f is continuous at c iff ( given any parameter x in the domain of f we have f(x) ≈ f(c) as x → c )."

One could define "f has limit L near c iff ( given any parameter x in the domain of f we have f(x) ≈ L as x → c )."

Both of these could be interpreted via the standard δ-ε formulation, or via sequences, or simply in an axiomatic framework.

For example, here's the sequence formulation:

"f is continuous at c iff ( given any sequence x from the domain of f we have f(x[n]) ≈ f(c) as natural n → ∞ )."

If it looks too similar, it's deliberate. Of course some people write things like "lim[n→∞] f(x[n]) = f(c)", but that's actually technically ambiguous because it doesn't specify what kind of entity n is.

But anyway, yes the point is that f is continuous at c iff f has limit f(c) at c.

Anyway I'm going off soon. See you both next time! =)

okay!