Nonconstant
A function is called nonconstant if it takes more than one value (if there is more than one element in its range). For example, the polynomial with the real numbers as domain and codomain is nonconstant. We can show this simply by noting that
and
, so the function takes at least two different values. However, the function
such that
for all
is a constant function, as the co-domain of the function remains the same regardless of changes to the domain.
Note that recognizing non-constant functions is not always trivial. For example, the function which takes an integer
, computes the value of
and then takes the remainder of this number on division by 3 appears quite complicated but turns out to be identical to the last function in the previous paragraph: it only takes the value 1.