In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that point and by the same number.
Locally bounded function A real-valued or complex-valued function f {\displaystyle f} defined on some topological space X {\displaystyle X} is called a locally bounded functional if for any x 0 ∈ X {\displaystyle x_{0}\in X} there exists a neighborhood A {\displaystyle A} of x 0 {\displaystyle x_{0}} such that f ( A ) {\displaystyle f(A)} is a bounded set. That is, for some number M > 0 {\displaystyle M>0} one has
| f ( x ) | ≤ M for all x ∈ A . {\displaystyle |f(x)|\leq M\quad {\text{ for all }}x\in A.}
In other words, for each x {\displaystyle x} one can find a constant, depending on x , {\displaystyle x,} which is larger than all the values of the function in the neighborhood of x . {\displaystyle x.} Compare this with a bounded function, for which the constant does not depend on x . {\displaystyle x.} Obviously, if a function is bounded then it is locally bounded. The converse is not true in general (see below). This definition can be extended to the case when f : X → Y {\displaystyle f:X\to Y} takes values in some metric space ( Y , d ) . {\displaystyle (Y,d).} Then the inequality above needs to be replaced with
d ( f ( x ) , y ) ≤ M for all x ∈ A , {\displaystyle d(f(x),y)\leq M\quad {\text{ for all }}x\in A,}
where y ∈ Y {\displaystyle y\in Y} is some point in the metric space. The choice of y {\displaystyle y} does not affect the definition; choosing a different y {\displaystyle y} will at most increase the constant r {\displaystyle r} for which this inequality is true.
Examples The function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by f ( x ) = 1 x 2 + 1 {\displaystyle f(x)={\frac {1}{x^{2}+1}}} is bounded, because 0 ≤ f ( x ) ≤ 1 {\displaystyle 0\leq f(x)\leq 1} for all x . {\displaystyle x.} Therefore, it is also locally bounded. The function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by f ( x ) = 2 x + 3 {\displaystyle f(x)=2x+3} is not bounded, as it becomes arbitrarily large. However, it is locally bounded because for each a , {\displaystyle a,} | f ( x ) | ≤ M {\displaystyle |f(x)|\leq M} in the neighborhood ( a − 1 , a + 1 ) , {\displaystyle (a-1,a+1),} where M = 2 | a | + 5. {\displaystyle M=2|a|+5.}
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