* By definition, <math>f</math> said to be continuous at the origin if for every open (or closed) ball <math>B_r</math> of radius <math>r > 0</math> centered at <math>0</math> in the codomain <math>\mathbb{F},</math> there exists some [[Neighbourhood (mathematics)|neighborhood]] <math>U</math> of the origin in <math>X</math> such that <math>f(U) \subseteq B_r.</math> * If <math>B_r</math> is a closed ball then the condition <math>f(U) \subseteq B_r</math> holds if and only if <math>\sup_{u \in U} |f(u)| \leq r.</math>
** However,It assumingis important <math>B_r</math> is a closed ball in this characterization. Assuming that <math>B_r</math> is instead an open ball, then <math>\sup_{u \in U} |f(u)| < r</math> is a sufficient but {{em|not necessary}} condition for <math>f(U) \subseteq B_r</math> to be true (consider for example when <math>f = \operatorname{Id}</math> is the identity map on <math>X = \mathbb{F}</math> and <math>U = B_r</math>), whereas the non-strict inequality <math>\sup_{u \in U} |f(u)| \leq r</math> is instead a necessary but {{em|not sufficient}} condition for <math>f(U) \subseteq B_r</math> to be true (consider for example <math>X = \R, f = \operatorname{Id},</math> and the closed neighborhood <math>U = [-r, r]</math>). This is one of several reasons why many definitions involving linear functionals, such as [[polar set]]s for example, involve closed (rather than open) neighborhoods and non-strict <math>\,\leq\,</math> (rather than strict<math>\,<\,</math>) inequalities.
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<li><math>f</math> is [[#bounded on a neighborhood|bounded on a neighborhood]] (of some point). Said differently, <math>f</math> is a [[#locally bounded at|locally bounded at some point]] of its ___domain.
