What are the limitations of computer plotting?

Question

The basic way of representing a function $f: \mathbb{R}\rightarrow \mathbb{R}$ with a computer, crudely put, is "connecting a number of dots". We can improve the quality of the representation by increasing the number of dots.

While this method is great for the most usual functions, specially continuous or piecewise continous functions, there are cases where it isn't. As in numerical analysis, the most obvious issue with this approach is numeric precission, but I can think of a few other issues derived from irrationals and essential discontinuities. Maybe there is something such as "symbolic plotting", akin to "symbolic computation", but I haven't found any reference to it.

Is there a comprehensive list of these issues, or a paper caracterizing "plottable" functions?

Answer 1

See these papers and the references therein:

• Honest plotting, global extrema, and interval arithmetic, by Richard Fateman

• From honest to intelligent plotting by Ron Avitzur et al

• Efficient plotting the functions with discontinuities based on combined sampling, by Tomáš Bayer

• Reliable two-dimensional graphing methods for mathematical formulae with two free variables, by Jeff Tupper

Answer 2

My attempt: a function is said $\epsilon-$plottable if every point of its graph falls within a radius $\epsilon$ of some $(\hat x_i,\hat y_i)$, where $\hat x_i$ and $\hat y_i$ are numbers that admit a finite representation (such as, but not necessarily, floating-point), and no $(\hat x_i,\hat y_i)$ has an empty $\epsilon-$neighborhoohd.

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This definition is not satisfactory, as it does not account for the fact that $\hat x,\hat y$ should be related by an equation similar to $y=f(x)$. But numerically, we don't have access to $f(x)$, and not even $f(\hat x)$.

Also, if $\epsilon$ exceeds the numerical resolution, by this definition any function is plottable.