Constant Function¶

Definition¶

A constant function is a function whose (output) value is the same for every input value, expressed as $f(x) = c\text{,}$ where $c$ is any fixed number (a constant). Equivalently, a constant function is a function whose range is a singleton set.

Interpretation¶

A constant function ignores its input: every $x$ in the domain gives the same output $c$. It is the simplest function there is and the baseline case in many results.

Its graph over an interval is the horizontal line $y=c$$c$, and its range is the single value $c$. It is bounded, and it is both increasing and decreasing in the weak sense of a monotone function, since $f(x)\le f(y)$ and $f(x)\ge f(y)$ both hold.

A common slip is to think that $f(x)=c$ is not really a function because $x$ does not appear in the formula. A function only has to assign one output to each input, and here it assigns $c$ to every input.

Key Results¶

• The graph of a constant function (defined on an interval) is a horizontal line.
• Constant functions have zero derivatives at any point where the derivative exists.

Examples¶

• The zero function $f(x)=0$ is a constant function.

Prerequisites¶

• Function
• Singleton Set

Related Concepts¶

• Differentiation Rules

Sources¶

• Math I: Calculus · Lecture 4

Last updated: 20 October 2026 03:02