For a visualization for educational purposes I'd like to construct a function $f$ that interpolates a set of points $(x_i,y_i)$ for $i=1,\ldots,N$ where $x_i < x_{i+1}$ are all distinct points on the interval $[0,1]$ and $y_i > 0$ for all $i$, and additionally $f$ should be "as smooth as possible" (at least continuous perferably $C^1$ or smoother).
So far this could easily be implemented e.g. using splines. But there two additional criteria that I need:
- The function $F(x) := \int_0^x f(t)dt$ should also be easy to evaluate and
- its inverse $F^{-1}(y)$ must also be easy to evaluate. (The inverse $F^{-1}$ of $F$ exists, because $f$ should be positive (as all the $y_i$ are positive) and $F$ therefore is monotonic.)
For a little bit more context: The function $f$ would be used as a probability distribution function that you can easily modify by manipulating the $y_i$.