Decision tree functions are discontinuous functions of the predictors. Have continuous decision trees with smooth transitions been studied? For example, a decision tree in two variables
f(x1,x2) = c1 if x1 < t1 and x2 < t2 c2 if x1 < t1 and x2 >= t2 c3 if x1 >= t1 and x2 < t3 c4 if x1 >= t1 and x2 >= t3
could be replaced by
f(x1,x2) = c1*lg(t1-x1)*lg(t2-x2) + c2*lg(t1-x1)*lg(x2-t2) + c3*lg(x1-t1)*lg(t3-x2) + c4*lg(x1-t1)*lg(x2-t3)
where lg is the logistic function.