8.25 min read

Constant Functions

The simplest scalar-valued function is a constant function: it outputs the same number $c$ regardless of the input $\mathbf{x}$ . Written as $f(\mathbf{x}) = c$ for all $\mathbf{x} \in \mathbb{R}^n$ . The graph of a constant function is a flat horizontal hyperplane at height $c$ .

Constant functions are degree 0 — no variables appear at all. They serve as the baseline case when classifying functions by their polynomial degree. A function that looks like $f(\mathbf{x}) = 5$ is constant; a function like $f(\mathbf{x}) = 5 + x_1$ is not, because it changes as $x_1$ changes.

Formal View

Definition 8.2 — Constant Function

A function

f: \mathbb{R}^n \to \mathbb{R}

is constant if there exists

c \in \mathbb{R}

such that

f(\mathbf{x}) = c

for all

\mathbf{x} \in \mathbb{R}^n

. It has polynomial degree 0.

Why This Matters

Constant functions appear as trivial edge cases in classification theorems and as offsets in affine functions.

A constant bias term $b$ in a neural network layer: $f(\mathbf{x}) = \mathbf{w}^\top \mathbf{x} + b$ has both a linear part and a constant part.
Regularization penalties that add a fixed constant to a loss function.
Lower bounds in optimization: proving a cost function is at least $c > 0$ everywhere.

Learning Resources

Types of functions — constant, linear, quadratic

Khan Academy

Quick overview distinguishing constant, linear, and quadratic functions.

6 min

Quiz

Question 1

The function $f(x_1, x_2) = 7$ is a constant function on $\mathbb{R}^2$ .

Question 2

What is the polynomial degree of a constant function?

Common Mistakes

Calling $f(\mathbf{x}) = c + x_1$ a "constant" function — only the $c$ part is constant; the function itself varies with $x_1$ .
Confusing a constant function with a zero function — a constant function outputs $c$ which may be nonzero, while the zero function outputs 0.