Your statment:
'Interpolation assumes smoothness & continuity between consecutive data points when connected using curves/lines'
No, at least not in general.
Linear problem (most probably not what you are looking for):
In case your task is related to a linear problem, then you can try 'linear regression'.
This means all your data (x/y sample pairs) can be completely unorderd. Especally this means you can have several 'y' values for one and the same 'x' value.
Polynomial, whatever:
Here we come to a basic thing that we need first to realize what a mathematical function means. E.g.
Function (mathematics) - Wikipedia[
^]
E.g. a 2D function (at least a 'Continuous function') can never have more than one 'y' value for a given 'x' value. Otherwhise it is not a function in therm of math.