One final note, perhaps an alternative approach would be to change the original $f$ so that $f$ would be one-to-one by making it so that it would map to a continuous line instead of a staircase function. LabChart Reader is a free application that allows you, your colleagues and students to share, view and conduct limited analyses of LabChart data.
What it seems to me is that you have defined a function in the following way:
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