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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?

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    $\begingroup$ This "partial graph connectivity ==> continuity" claim is false. $\endgroup$
    – Wlod AA
    Commented Oct 21, 2022 at 4:57
  • $\begingroup$ This is false even when $A=B=N_a=\mathbb R$. $\endgroup$
    – Wlod AA
    Commented Oct 21, 2022 at 5:04
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    $\begingroup$ The "continuity" in [This "partial graph connectivity ==> continuity" claim is false.] is the usual continuity? $\endgroup$
    – mamediz
    Commented Oct 21, 2022 at 5:18
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    $\begingroup$ Oh, do you mean concepts that are somewhat similar but not equivalent to continuity? $\endgroup$
    – Wlod AA
    Commented Oct 21, 2022 at 5:26
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    $\begingroup$ Yes, that's exactly what I mean with this question. $\endgroup$
    – mamediz
    Commented Oct 21, 2022 at 11:15

1 Answer 1

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The function $f$, defined by $f(0)=0$ and $f(x)=\sin\frac1x$ if $x\neq0$, is not continuous at $0$, but its graph is connected over every open interval that contains $0$. With a bit of work, shifting $f$ around and scaling it, you can create a function that exhibits this behaviour at every rational number.

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    $\begingroup$ That was about the simplest of examples that I had in my mind when commenting on OP post. $\endgroup$
    – Wlod AA
    Commented Oct 21, 2022 at 9:06
  • $\begingroup$ There are functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(U)=\mathbb{R}$ for every nonempty open set $U$, so that's also a pretty discontinous counterexample $\endgroup$
    – Saúl RM
    Commented Oct 21, 2022 at 22:14
  • $\begingroup$ @WlodAA I do not see any examples in your comments. You just said it is fale when $A=B=N_a=\mathbb{R}$. That is not an example. $\endgroup$
    – Piotr Hajlasz
    Commented Oct 22, 2022 at 16:02
  • $\begingroup$ What you describe is essentially Sierpinski's "punctiform connected graph" where punctiform = no compact connected subset. The OP may want to see the dissertation here: apronus.com/static/MRWojcikPhD.pdf, which is a fairy complete study of this type of thing. $\endgroup$
    – D.S. Lipham
    Commented Oct 22, 2022 at 22:58
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    $\begingroup$ @PiotrHajlasz, I said in my mind. (Right, of course, u'r not a reader of my mind, and let's keep it this way). $\endgroup$
    – Wlod AA
    Commented Oct 22, 2022 at 23:25

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