# What are the conditions necessary and sufficient for a function $f : \mathbb{R} \to \mathbb{R}$ to be represented by a graph?

I am curious to know the answer for this question.

What are the conditions necessary and sufficient for a function $f : \mathbb{R} \to \mathbb{R}$ to be represented by a graph ?

For example we can represent any trignometric (sin,cos..etc) functions by plotting a graph on a paper.

some of my thoughts are

differentiability is one condition i can think of.. but i am not sure if it is required over the entire interval where the function is defined. Other things bothering me are finiteness of number of maxima and minima in the desired interval. Oscillating singularities should be definitely avoided.

Added : to be able to draw graph : I am not able to describe it mathematically, but it is like this. Given a pencil and a paper of sufficient dimension, one can draw the graph of the function in any closed interval in its domain, by using one's hand.

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I'm not quite sure what you mean by a graph - are you thinking, for example of a "continuous" locus having finite "length" with concepts of continuity and length to be defined? Any function of the kind you describe can be represented as a set of points in the Real plane: what makes that collection of points into a "graph"? –  Mark Bennet Sep 16 '11 at 15:35
@Mark Bennet : I am not able to describe it mathematically, but it is like this. Given a pencil and a paper of sufficient dimension, one can draw the graph of the function in any closed interval in its domain, by using one's hand. –  Rajesh D Sep 16 '11 at 16:00
I see, so you're excluding monsters like $\sin\frac1{x}$? –  Ｊ. Ｍ. Sep 16 '11 at 16:07
@J.M. : even of these type : en.wikipedia.org/wiki/Weierstrass_function –  Rajesh D Sep 16 '11 at 16:10
I'm not so sure this question is well-posed. Is $x^3 \sin{\frac{1}{x}}$ allowed or not? –  t.b. Sep 16 '11 at 16:25

I think the notion that best captures this is rectifiability -- see the Wikipedia article on arc length for a definition. Briefly, a curve is rectifiable if it has a well-defined, finite length.

Now, you certainly can't draw a graph of infinite size, so you must restrict yourself to a bounded interval $[a,b]$. I suggest that a function $f:[a,b]\rightarrow \mathbb R$ is 'graphable' if its graph {$(x,f(x)):x\in[a,b]$} consists of a finite number of rectifiable curves. Note that this allows for discontinuities.

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Thank you for the suggestion, this precisely answers Theo's question about functions of the type $x^3sin(\frac{1}{x})$. But rectifiable curve is not actually what i had in mind, and after seeing your answer, I'd like to add an interesting property to what constitutes as 'graphable' using a pencil and a paper. May i ask you whether i can do this by editing the same question or by asking another question ? –  Rajesh D Sep 16 '11 at 17:13

I assume by graph you're meaning the following:

The piece of paper has some dimensions are thus it's coordinates are within range x=[0..m], y=[0..n]. Then a point inside the paper is a pair (x,y). For it to be a graph, a point would be drawn black if a pair (x,f(x)) is the point and white otherwise. This has to happen for every possible x, within the allowed range. Every pair (x,y) is going to get a colour using this procedure.

Now a function f : R->R and graph has some properties that are important:

1. for every x : R, there exists only one number y : R so that y = f(x).
2. Thus the graph is a subset of the pair space RxR. Some points are black, and others are white, and the black points form a subset.
3. for all points (x,f(x)) to be inside the paper, all the pairs need to be within the correct range, giving us: x : [0..m] and f(x) : [0..n]. This further restricts which values of x are allowed. If f(x) is not within [0..n], then x must be removed from the allowed range [0..m] and it becomes [0..s] and [s..m].
4. for sin/cos/tan etc graphs would be allowed, you might need to scale and reposition the coordinate values. Otherwise negative values of f(x) would not be allowed. Scaling happens by [s,r] => [ks,kr] and [s,r] => [s+a,r+a].
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