function as a net? One can see net as a generalization of a sequence. 
As done in http://en.wikipedia.org/wiki/Net_(mathematics), in the special case where $f: M\backslash \{a\} \rightarrow X$ where M is a metric space and X is a topological space, one can see $f$ as a net from the directed set $M\backslash \{a\}$ with partial order given by the distance to $a\in M$. The closer a point is to $a$, the "greater" it is.
Question: It is possible to see an arbitrary continuous function from a topological space to another as a net?
(it's a question for free, i was not looking for any application of it)
Edit: actually one can define $f:M \rightarrow X$. it's just when we want to have a directed set that we take a point away.
 A: The conventional way (to treat a topological limit as a net limit) is like this.  (Taken from Kelley's paper of around 1950).  [Kelley, J. L. Convergence in topology. Duke Math. J. 17, (1950). 277–283.]
Let $M$, $X$ be topological spaces.  Let $f : M \to X$ be a function.  We are interested in $\lim_{m \to a} f(m)$.  Define a directed set $D$ as follows.  $D$ consists of pairs $(E,p)$, where $E$ is a neighborhood of $a$ and $p \in E.$  For ordering: $(E_1,p_1) \ge (E_2,p_2)$ iff $E_1 \subseteq E_2$.  Check it is directed.  Define a net $T_f : D \to X$ by $T_f(E,p) = f(p)$.  Then $x \in X$ is a value of $\lim_{m\to a} f(m)$ iff it is a limit of the net $T_f$.  
The usual modification can be made when $f$ is defined on a subset $N$ of $M$ and $a$ is in the closure of $N$.  Take pairs $(E,p)$ where $E$ is a neighborhood of $a$ and $p \in E \cap N$.
REMARK. by taking all points of every neighborhood, we avoid requiring the Axiom of Choice.
A: If you assume the Well-ordering theorem then yes! In that case, if we have $f:X\rightarrow  Y$ then the Well-ordering theorem gives us a well-ordering of $X$ hence it becomes a directed set.
It is still interesting to see if one can use the topological structure on $X$ to turn it into a directed set.
A: Yes, you can. A function is continuous iff it preserves convergence of nets. 
Unpacking this, we see that this means when given a function $f: X \rightarrow Y$, then whenever we are given a convergent net $u\rightarrow x$, we have that $fu \rightarrow fx$. 
It’s perhaps worth pointing out that there are several notions which are tied together here: that of a net, a filterbase and a filter. The latter are generalisations of the neighbourhood filter and filterbase of open sets. The notion of a net and filter are connected, by seeing that any net defines a net of opens by choosing an open set that includes that point (and none of the preceding points), and contrarywise, such a net of opens defines a net of points. 
In a sense, the net of opens is the more natural notion as it simply a morphism of directed sets. We then note, that the natural indexing set for such a net is the directed set of neighbourhoods and then this brings us to the notion of a filter. 
