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In contrast, a map <math>F : X \to Y</math> is said to be {{em|{{visible anchor|bounded on a neighborhood of}}}} a point <math>x \in X</math> or {{em|{{visible anchor|locally bounded at}}}} <math>x</math> if there exists a [[Neighborhood (mathematics)|neighborhood]] <math>U</math> of this point in <math>X</math> such that <math>F(U)</math> is a [[Bounded set (topological vector space)|bounded subset]] of <math>Y.</math>
If a linear map is locally bounded at {{em|some}} point then it is locally bounded at {{em|every}} point.
A map is called [[Locally bounded function|{{em|{{visible anchor|locally bounded}}}}]] if it is locally bounded at every point of its ___domain, but some functional analysis authors define "locally bounded" to be a synonym of "[[bounded linear operator]]", which are related but {{em|not}} equivalent concepts. For this reason, this article will avoid the term "locally bounded" and instead say "locally bounded at every point" (there is no disagreement about the definition of "locally bounded {{em|at a point}}").
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{{See also|Local boundedness}}
A linear map that is bounded on a neighborhood (of some/every point) is necessarily continuous{{sfn|Narici|Beckenstein|2011|pp=156-175}} (even if its ___domain is not a [[normed space]]). The next example shows that the converse is not always guaranteed.
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