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.

Solution 1:

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.