Restriction of a Function¶

Definition¶

Let $f\colon A\to B$ be a function and let $S\subseteq A$ $A$ . The restriction of $f$ to $S$ is the function

f|_S\colon S\to B, \qquad f|_S(x)=f(x).

Only the domain is changed; the rule used to compute the value is the same.

Interpretation¶

A restriction keeps the rule of $f$ and only shrinks its domain to a subset $S$ . This is the formal step behind optimizing a function subject to constraints: the objective stays the same, only the admissible inputs change.

Graph of f on the interval A from 0 to 6 drawn dashed, with the part over the subinterval S from c to d highlighted as the restriction of f to S.

The dashed curve is $f$ on $A=[0,6]$ $[0, 6]$ ; the red piece over $S=[c,d]$ is the graph of $f|_S$ , with the same values and a smaller domain. Its largest value is at the left endpoint $c$ , where $f$ itself is decreasing and has no local maximum. The extrema of $f|_S$ are the constrained extrema of $f$ , and they can sit where $f$ has none.

Other properties change with the domain too: $f|_S$ may be injective or bounded when $f$ is not: $x^2$ is injective on $[0,\infty)$ (see the Examples) and bounded on $[0,1]$ , but neither on $\mathbb{R}$ . In the same way, $\arcsin$ is defined as the inverse of the restriction of $\sin$ to $[-\pi/2,\pi/2]$ (inverse trigonometric functions), and the multivariable inverse function theorem inverts $\boldsymbol{f}|_V$ on a small neighbourhood $V$ . For an equality constraint $g(\boldsymbol{x})=b$ $b$ , the restricted domain is a level set of $g$ .

A common slip is to look for the extrema of $f|_S$ among the stationary points of $f$ . On the unit circle, $f(x,y)=x+y$ is largest at $(1/\sqrt{2},1/\sqrt{2})$ , although its gradient $(1,1)$ is never zero.

Key Results¶

The restricted function can have extrema that the unrestricted function does not have.
A constraint set often defines the domain of the restricted objective.
Properties such as injectivity can change after restriction.

Examples¶

The function $f(x,y)=x+y$ restricted to the circle $x^2+y^2=1$ becomes an objective on that circle only.
The function $f(x)=x^2$ is not injective on $\mathbb{R}$ , but its restriction to $[0,\infty)$ is injective.

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 3, Lecture 1

Aliases: restriction to a set Keywords: function restriction, restricted function Symbols: infinity, pi, real numbers, square root, subset

Last updated: 27 October 2026 11:52