What are the limitations of computer plotting?

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.

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