According to definition of Interpolation -
A function $y=P(x)$ can interpolate a set of data points if $y_i = P(x_i) | 1\le i\le n$ for the set of data points being - $(x_1,y_1), ..... , (x_n,y_n)$.
This means that we can interpolate a curve only when it can be represented by a function. But what if a curve cannot be represented by a function, so can we represent it by a step-wise interpolation?
Easy solution: use two functions $P$, $Q$: $$P(i) = x_i,\qquad Q(i) = y_i,\qquad 1\le i\le n.$$ The image set of $[1,n]$ by $(P,Q)$ is the representation that you are searching.