# Is step-wise interpolation possible for a curve which cannot be represented by a function?

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?

• What means a "curve that cannot be represented by a function"? – Martín-Blas Pérez Pinilla Mar 3 '18 at 12:18
• @Martín-BlasPérezPinilla consider an "S" shaped curve - It cannot be represented by any function , because you can find x such that it is related with 2 y's – Aashish Loknath Panigrahi Mar 3 '18 at 16:08
• The correct denomination: "curve that isn't the graph of a function." – Martín-Blas Pérez Pinilla Mar 3 '18 at 16:12

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.