does any random curve point to a function?

Question

If i draw any random smooth curve on a graph will it correspond to a function ie. infinite series or so of polynomial or fourier series the curve is differentiable and integrable at all points so it may be possible, am i right?

Answer 1

Any set of points in the plane (curves are sets of points) such that for each vertical line there is exactly one point from your set on that line, will correspond to a function on the real numbers (that set is the graph of that function). No other sets correspond to functions on the real numbers.