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<li>There exists some neighborhood <math>U</math> of the origin such that <math>\sup_{u \in U} |f(u)| \leq 1</math>
* This inequality holds if and only if <math>\sup_{x \in r U} |f(x)| \leq r</math> for every real <math>r > 0,</math> which shows that the positive scalar multiples <math>\{r U : r > 0\}</math> of this single neighborhood <math>U</math> will satisfy the definition of [[Continuity at a point|continuity at the origin]] given in (4) above.
* By definition of the set <math>U^\circ,</math> which is called the [[Polar set|(absolute) polar]] of <math>U,</math> the inequality <math>\sup_{u \in U} |f(u)| \leq 1</math> holds if and only if <math>f \in U^\circ.</math> Polar sets, and
</li>
<li><math>f</math> is a [[#locally bounded|locally bounded at every point]] of its ___domain.</li>
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