Is it possible to get the function's expression through its graph/curve?

Question

We all know how to draw the curve of a function through its expression but is it possible to reverse this process?, e.g. I will give you a detailed curve (of a non-linear a function) and you find the expression related to this function, since each curve has only a single function as I know.

Any help would be appreciated, thanks in advance.

Answer 1

Given a function $f:\mathbb{R}\to\mathbb{R}$, it is easy to draw its graph $G$: we simply draw all points $(x, f(x))$.

Given a graph $G$, consisting of points $(x, y)$, we can easily define a function $f$: the value of the function in point $x$ is the only $y$ that appears in $G$ in a pair with $x$.

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The second procedure seems to be much more implicit, but is it really?

1. Can you really simply draw all the points $(x, f(x))$?
2. In the second case, can you really explicitly define $G$ without using $f$ as a part of definition? That is, can you really give a "a detailed curve" (these are your words) without using the $f$ as a part of definition?

IMHO, the answer to the both questions is no, because

1. You can not draw infinitely many points, or even a single point, with an infinite precision.
2. I cannot think of a way to define an infinite set of points $G$ which would satisfy the property if $(x, y_1), (x, y_2)\in G$, then $y_1 = y_2$, without using the function itself.