Suppose I have an ***unknown*** function $f(x)$ that is monotonic, convex and nonlinear by definition. I somehow evaluated it at 1000 nonuniform $x$ points. By ploting these data, I found a very obvious nonlinear curve with a dozen of outliers, which, e.g., contain sudden small or large peaks and valleys and violate monotonicity and convexity. How to carry out outlier recognition automatically? Sth. more systematic than inspecting by eyes...

Note that my ultimate purpose is not finding any fit of these data and I don't want any modification among the data except excluding outliers.

The data set is (***updated***, more reasonable, different from the one plotted in the comment below, apology for that I cannot retain the original one due to character limit)

```
data = {0,197.1278166},{3.356150145 10^-5,197.1004078},{6.745694819 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