I gave a take home test to my numerical analysis students. This was one of the problems (a “warm up”):
I wanted them to choose a technique (say, a Lagrange polynomial or a cubic spline) to approximate the function at those values.
Most did fine. However one student’s answer bothered me: the student used a proper method but then rounded…to an integer each time.
Yes, in this special case, by pure chance..this turned out to be correct as this data comes from the gamma function.
But in general, that is terrible intuition. There is no reason a function should take integers to integers, even if that always happens with polynomials with integer coefficients.
Unfortunately, too many elementary calculus textbooks and too many math educators reinforce such bad intuition.