I remember once reading that "a continuous function can be loosely described as a function whose graph can be drawn without lifting the pen from the paper". We all know that this is not true. I was looking for a way to give a well defined concept of a continuous function from this idea, maybe:

Definition: Let $f : A \to B$ a function and $a \in A$. We say that $f$ is continuous at $a$ if there is a neighborhood $N_a$ of $a$ such that the graph of $f |_{N_a}$ is connected.

Let's say that we restrict $A$ and $B$ to be locally connected topological spaces.

What can be done to improve this definition? What would be the consequences if we develop a theory with a concept like this?

I remember once reading that "a continuous function can be loosely described as a function whose graph can be drawn without lifting the pen from the paper". We all know that this is not true. I was looking for a way to give a well defined concept of a "somewhat continuous" function from this idea. Of course it would be a totally different definition and not at all equivalent to continuity as we know, to not cause any confusion let's call it p2p and avoid the use of "continuous".

maybe:

Definition: Let $f : A \to B$ a function and $a \in A$. We say that $f$ is p2p at $a$ if there is a neighborhood $N_a$ of $a$ such that the graph of $f |_{N_a}$ is connected.

Let's say that we restrict $A$ and $B$ to be locally connected topological spaces.

What can be done to improve this definition? What would be the consequences if we develop a theory with a concept like this in place of the usual continuity?

In summary, my question is, what is the best we can do to give a well defined concept of a "somewhat continuous" function that capture the physical idea of a pencil-to-paper graph?