Given $f: \mathbb R \to \mathbb R$, what can we say about its plot (that is, about what a sketch of its graph may look like)? Continuous functions are often described as "functions we can draw the graph of" - can we make this more precise? What about discontinuous functions like $x \mapsto \sin (1/x)$ or the Dirichlet function? Or pathological functions such as those dense in the plane?
An initial attempt:
Given a function, define its graph $G$ as $G(f) = \{(x, f(x)) : x \in \mathbb R\}$. Given a graph $G \subseteq \mathbb R^2$, then $P \subseteq \mathbb R^2$ is a plot of $G$ if both the following conditions are met:
- $G \subseteq P$
- $P$ is the arbitrary union of open balls, each of which: a. Has a constant diameter $w$ (called the pen-width of $P$) b. Includes at least one point from $G$ (that is, $G \cap B \neq \emptyset$).
Consequently:
- Most functions can have many plots of the same pen-width. Question: Are there functions that for a given pen-width admit only one plot?
- The plot of a continuous function seems to match our intuition
- The plot of the Dirichlet function is equal to $P(G(x \mapsto 0)) \cup P(G(x \mapsto 1))$, as we'd expect. ($P$ is not a function but is used similar to big-$O$ notation.)
- The plot of a function dense in the plane is, at any pen-width, the plane itself
- I do not know how to characterize the plot of $x \mapsto \sin(1/x)$ or $x \mapsto x \sin(1/x)$
- Given a plot, can we always tell if it is the plot of a continuous function? For what pen-widths do discontinuous functions take on continuous plots? What is the relationship between plots, continuity, and differentiability?
- Is there a way to tighten the definition of plot so that for a given pen-width a function has only one plot?
- Does the definition work well for higher dimensions and other metric spaces?
This question is not limited to those questions: Any reasonable definition of a plot of a function or discussion of its properties is responsive.