http://debianws.lexgopc.com/wiki143/index.php?action=history&feed=atom&title=Symmetrically_continuous_functionSymmetrically continuous function - Revision history2026-10-25T14:07:00ZRevision history for this page on the wikiMediaWiki 1.43.1http://debianws.lexgopc.com/wiki143/index.php?title=Symmetrically_continuous_function&diff=2200804&oldid=prev193.205.210.82: /* top */added some insight2023-03-08T09:09:03Z<p><span class="autocomment">top: </span>added some insight</p>
<p><b>New page</b></p><div>In [[mathematics]], a function <math>f: \mathbb{R} \to \mathbb{R} </math> is '''symmetrically continuous''' at a point ''x'' if<br />
:<math>\lim_{h\to 0} f(x+h)-f(x-h) = 0.</math><br />
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The usual definition of [[continuous function|continuity]] implies symmetric continuity, but the converse is not true. For example, the function <math>x^{-2}</math> is symmetrically continuous at <math>x=0</math>, but not continuous.<br />
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Also, [[Symmetric derivative|symmetric differentiability]] implies symmetric continuity, but the converse is not true just like usual continuity does not imply differentiability.<br />
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The set of the symmetrically continuous functions, with the usual [[scalar multiplication]] can be easily shown to have the structure of a [[vector space]] over <math>\mathbb{R}</math>, similarly to the usually continuous functions, which form a [[linear subspace]] within it.<br />
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== References ==<br />
* {{cite book<br />
| first= Brian S. <br />
| last= Thomson <br />
| year= 1994 <br />
| title= Symmetric Properties of Real Functions<br />
| publisher= Marcel Dekker<br />
| isbn= 0-8247-9230-0<br />
}}<br />
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[[Category:Differential calculus]]<br />
[[Category:Theory of continuous functions]]<br />
[[Category:Types of functions]]<br />
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{{mathanalysis-stub}}</div>193.205.210.82