 A slightly more complicated example isformula_3"f" is still intrinsic one-dimensional, which can be seen by making a variable transformation

formula_4

formula_5

which gives

formula_6

Since the variation in "f" can be described by the single variable "y1" its intrinsic dimension is one.For the case that "f" is constant, its intrinsic dimension is zero since no variable is needed to describe variation.For the general case, when the intrinsic dimension of the two-variable function "f" is neither zero or one, it is two.