@Jim Baldwin I think that may be exactly what I was looking for! Many thanks to you and @Neil Singer. You don't know how much I appreciate it!

I've tried Interpolation and InterpolatingPolynomial. They both work well, but are not quite what I need. I've attached what I hope to achieve. An input of 2d points and an output of a pw function consisting only of linear pieces. So yes, I suppose I am looking for analytical expressions for simple curves. Obviously one could do these by hand but I'd prefer something that does it for me in seconds.

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Understood. But I would have to believe that this can't be related to "real data" in that all of the data points in your example consist of integers. Do your sample datasets involve numbers other than integers? (This is part of my question about being real specific.)

Hello Jim, I'm looking specifically for a piecewise composed solely of linear functions. If my input is n amount of points, my hope is for it to return a pw function with n-1 pieces (or less if more than three consecutive points are colinear though that's not a necessity). My reply to "Neil" has an example of what I'm hoping for. As far as regressions, I'm trying to avoid having to use them though I appreciate the suggestion.

Do you have any restrictions/knowledge about the form that the pieces of the piecewise function can take and (at least roughly) the number pieces? For example, are you looking for a parsimonious number of connected linear or quadratic fits?

If you have no idea about what to expect (and that's not meant to be a slam), then some sort of nonparametric regression might be what you want. Maybe a gam (generalized additive model) or @AntonAntonov 's quantile regression package.

If you could expand on the structure of these datasets, then you would likely get more targeted help.