Are general topology and real analysis school definitions of limit equivalent? I am currently facing an issue which I should have resolved ages ago.
I can't find my mistake in this.
On one hand, a function $f : X \rightarrow Y$ between two topological spaces is said to have limit $l \in Y$ at point $x \in X$ if for any neighbourhood $N \subset Y$ of $l$ the preimage $f^{-1}(N)\subset X$ of $N$ is a neighbourhood of $x$.
 But then this actually means that if Y is Hausdorff and $f$ has a limit at $x \in X$, then $f(x)$ is that limit. Indeed let $l=\lim_{t \rightarrow x} f(t)$. Then $x$ is in $\cap_{N\in \mathcal{I}}f^{-1}(N)$ where $\mathcal I$ denotes the set of neighbourhoods of $l$. But because $Y$ is separated, $\cap_{N\in \mathcal{I}}f^{-1}(N)=\lbrace l \rbrace$, therefore $f(x)=l$.
Therefore, defining continuity at $x$ by "$\lim_{t \rightarrow x}f(t)$ exists and equals $f(x)$" should be replaced by "$\lim_{t \rightarrow x}f(t)$ exists", since I've just proven that if the limit exists, then it equals the value of the function at that point. But I know that all this is wrong somehow, because the function which assigns $1$ to $0$ and $0$ to any other real is said to have limit $0$ at $0$ but with what I've just said, it wouldn't be. So there is some inconsistency in all of this, but where ?
 A: I have found the answer to my own question so I'm posting it here.
My proof fails at the very start because even though the rest of it is correct, my definition of the limit at a point is wrong.
Wikipedia says that $f : X \rightarrow Y$ tends to $y \in Y$ at $x \in X$ if for every neighbourhood $N$ of $y$ there exists a neighbourhood $M$ of $x$ such that $f(M \setminus \lbrace x \rbrace ) \subset N$. In real analysis, this is equivalent to
$\forall \varepsilon > 0 \quad \exists \delta > 0 \quad | \quad 0 < |x-t| < \delta \quad \implies |f(t) - y| < \varepsilon   $
These definitions do allow limits at specific points to be different from the values at those same points.
A: The definitions are equivalent.   When you have a metric as you do with the reals the open sets still follow all the usual rules for a topological space. Maybe better than "equivalent" would be "consistent".  For one thing,  after all,  not all topological spaces have a metric,  or are metrizable. .. 
For the moment  I am ignoring the deeper questions of consistency that Gödel and other set theorists grapple with... But maybe you should investigate these...
