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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?

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  • $\begingroup$ What means a "curve that cannot be represented by a function"? $\endgroup$ Commented Mar 3, 2018 at 12:18
  • $\begingroup$ @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 $\endgroup$ Commented Mar 3, 2018 at 16:08
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    $\begingroup$ The correct denomination: "curve that isn't the graph of a function." $\endgroup$ Commented Mar 3, 2018 at 16:12

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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.

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