Why is pointwise continuity not useful in a general topological space? 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.
 A: 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 $x$.
A: In a metric space we begin with the $\epsilon$-$\delta$ definition of continuity, which is then globalized by the requirement that the $\epsilon$-$\delta$ holds at every point. This is a natural starting point because from the quantitative $\epsilon$-$\delta$ relation we are led to several related important concepts: 


*

*uniform continuity

*Hölder continuity

*Lipschitz continuity


None of the above related concepts exist in a topological space, where the notion of continuity is not quantifiable. 
To define continuity in  a topological space we operate not with a pair of nearby points $x_1,x_2$, but with a set of points; thus, the global picture emerges at once. "Preimage of an open set is open", a standard way to define continuity in a topological space, is naturally global. Equivalently, one can state it as "preimage of a closed set is closed". From here, your definition of continuity at a point,
$$A \in \mathcal{P}(X), \, x_0\in \overline{A}\implies f(x_0)\in \overline{f(A)}\tag{1}$$
is obtained by localizing a global definition : you have localized the property of "containing all its limit points" by focusing on a particular limit point. As a result, (1) is  more contrived than the global definition. If you try to reprove the basic results of point-set topology    always using (1) as definition of continuity, you will likely find that the proofs become more cumbersome.

I prefer to read "not such a useful"   as "not such a natural"; which however amounts to the same thing. 
