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 $p(x) = x^2 - x + 1$ with the real numbers as domain and codomain is nonconstant. We can show this simply by noting that $p(1) = 1$ and $p(2) = 3$, so the function takes at least two different values. However, the function $f: \mathbb{Z} \to \mathbb{Z}$ such that $f(x) = 1$ for all $x$ is a constant function, as the co-domain of the function remains the same.

Note that recognizing non-constant functions is not always trivial. For example, the function $f: \mathbb{Z} \to \mathbb{Z}$ which takes an integer $x$, computes the value of $x^5 -2x^4 -2x^3 - x^2 + x + 4$ 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.

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