TITLE: Why is pointwise continuity not useful in a general topological space?
QUESTION [6 upvotes]: On page 27 of Lee's Introduction to Topological Manifolds, he writes

In metric spaces, one usually first defines what it means to be continuous at a point...in topological spaces, continuity at a point is not such a useful
  concept.

Why not? We can define continuity at a point $x_0$ by the requirement that for all $A \in \mathcal{P}(X), \, x_0\in \overline{A}\implies f(x_0)\in \overline{f(A)}.$
I suppose it's not easy to say why something is not useful, but if anyone has some insight, I'd be glad to hear it.

REPLY [3 votes]: One possible characterization of "continuity of $f:X\to Y$ at a point $x\in X$" for $X,Y$ arbitrary topological spaces is that $$ \tag{1} \text{for every net $x_\alpha \to x$, we have $f(x_{\alpha}) \to f(x)$.}$$ 
Suppose we define a neighborhood of $x$ as a set containing an open set containing $x$. I claim that an equivalent condition to (1) is 
$$\tag{1'} \text{for every neighborhood $V$ of $f(x)$, $f^{-1}(V)$ is a neighborhood of $x$}.$$
I claim further that (1) is equivalent to
$$\tag{1''}\text{for every open set $V$ containing $f(x)$, $f^{-1}(V)$ is open}.$$
Therefore, equivalent to continuity is that (1') holds at all points $x$.