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- We say that a real function f is bounded from above if there is a number K such that for all x from the domain D (f) one has f (x) ≤ K. We say that a real function f is bounded if it is bounded both from above and below.
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If is real-valued and () for all in , then the function is said to be bounded (from) above by . If f ( x ) ≥ B {\displaystyle f(x)\geq B} for all x {\displaystyle x} in X {\displaystyle X} , then the function is said to be bounded (from) below by B {\displaystyle B} .
We say that a real function f is bounded if it is bounded both from above and below. Equivalently, a function f is bounded if there is a number h such that for all x from the domain D ( f ) one has - h ≤ f ( x ) ≤ h , that is, | f ( x )| ≤ h .
5 days ago · A set is said to be bounded from above if it has an upper bound. Consider the real numbers with their usual order. Then for any set M subset= R, the supremum supM exists (in R) if and only if M is bounded from above and nonempty.
- Related Notions
- Examples
- Boundedness Theorem
Weaker than boundedness is local boundedness. A family of bounded functions may be uniformly bounded. A bounded operator T : X → Y is not a bounded function in the sense of this page's definition (unless T = 0), but has the weaker property of preserving boundedness: Bounded sets M ⊆ X are mapped to bounded sets T(M) ⊆ Y. This definition can be exte...
The sine function sin : R → R is bounded since | sin ( x ) | ≤ 1 {\displaystyle |\sin(x)|\leq 1} for all x ∈ R {\displaystyle x\in \mathbf {R} } .
Recall that a function f {\displaystyle f} is bounded on a set A {\displaystyle A} if for every M ∈ R {\displaystyle M\in \mathbb {R} } , M > 0 {\displaystyle M>0} , then ∀ x ∈ A {\displaystyle \forall x\in A} , we have that ∣ f ( x ) ∣< M {\displaystyle \mid f(x)\mid
A is bounded above (or right bounded) iff there is q ∈ F such that. (∀x ∈ A) x ≤ q. In this case, p and q are called, respectively, a lower (or left) bound and an upper (or right) bound, of A. If both exist, we simply say that A is bounded (by p and q).
A set A ∈ ℝ of real numbers is bounded from above if there exists a real number M ∈ R, called an upper bound of A, such that x ≤ M for every x ∈ A (Hunter, n.d.). Basically, the above definition is saying there’s a real number, M, that we’ll call an upper bound.
Feb 10, 2013 · Bounded means bounded above and below. The function −x2 − x 2 is not bounded. In most applications to algorithms, it doesn't matter, since the functions are naturally non-negative, so are bounded below by 0 0. But in principle one should be careful.
