# Criteria for a function to be plottable

Assume that $f$ is a real function. My question is how can one decide if $f$ is plottable or not? My assumption is that $f$ must be of class $C^1$, but I am not aware of such a result. My assumption is based on the fact that there are differentiable functions that can not be plotted (e.g., $$f:\mathbb{R}\rightarrow\mathbb{R},\quad f(x)=\left\{ \begin{array} [c]{l}% x^{2}\sin\dfrac{1}{x},~\text{if }x\neq0\\ 0,~\text{if }x=0 \end{array} \right.$$

is not plottable in the neighborhood of $0$, since it's derivative does not have limit at $0$, hence the slope of the tangent to the graph varies in an "uncontrolable" fashion.

Edit: The problem is that I don't have an "official" definition for what "plottable" means, I only have an intuitive understanding of the concept. Also, it is clear that there are plotable functions which have points where they are not differentiable, e.g., $|x|$, so my requirement of being $C^1$ is not valid. Maybe, i should have said "picewise $C^1$".

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Can you please define plottable function? – Git Gud May 29 '13 at 20:06
Surely you can plot $|x|$, which is not $C^1$. – Pedro Tamaroff May 29 '13 at 20:06
Are you asking for functions whose graph forms a rectifiable curve, i.e. with finite arc length? – Rahul May 29 '13 at 20:09
I edited the post to answer the questions so far. – digital-Ink May 29 '13 at 20:15
@digital-Ink My take on this is: as you said, plottable is just an intuitive term. Since it is something you actually do or create in the real world, it can't possibily be treated mathematically. That being said, this turns into philosophical question rather than a mathematical one. In this context I'd say you need the function to be continuous except on at most a finite number of points. Also mathematical ideas often rise from trying to model reality and then you can define plottable function mathematically, but only in the realm of mathematics which you're using to model reality. – Git Gud May 29 '13 at 20:19

Without a precise definition of plottable, we cannot give a precise classification of the functions that are. Some things that can get in the way:
1) Too many discontinuities: The Dirichlet function, $1$ on the rationals and $0$ on the irrationals is an example. Roughly speaking, I would say we can handle a finite number of discontinuities as long as they are not to close together
2) Too much local variation: Even $\sin \frac 1x$ on $[\frac 1{100000},1]$ has so much variation that it will be a blur. This one is $C^{\infty}$
3) To much variation of scale: Think of taking a square wave with amplitude $1000$ and adding $\sin x$ to it. Of course we can round off the corners to get continuity. But to see what is going on, you need to see the square wave, and then the sine wave will be too small to see. You can have similar problems in the $x$ direction. Maybe you have stuff going on close to the origin, then around $x=100000$.
4) As a takeoff of 3, too much detail. There are only so many pixels on a piece of paper. If there is too much important stuff, some will get lost. Think of the Mandelbrot set (not a function but it gives the idea)
5) Too much global variation. What do you do with $e^{1000x}\sin x?$

I think the real answer is that almost no functions are plottable, but the ones we use are selected to usually be plottable. This is like the fact that almost every function is discontinuous, but most of the ones you see in class will be continuous.

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