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:The statement is not only correct, but it remains true if "the interior of the ___domain" is replaced by "in the ___domain". In fact, the definition of a limit is "For every positive real number {{math|''ε'' > 0}}, there is a positive real number {{math|''δ'' > 0}} such that <math>|f(\boldsymbol{x}) - L| < \varepsilon </math> for all {{math|'''''x'''''}} in the ___domain such that <math>d(\boldsymbol{x}, \boldsymbol{a})< \delta.</math>" If one takes {{math|1='''''x''''' = '''''a'''''}}, one has <math>d(\boldsymbol{x}, \boldsymbol{a})< \delta.</math> Thus the existence of a limit implies that <math>|f(\boldsymbol{a}) - L| < \varepsilon</math> for every {{math|''ε'' > 0}}. This implies <math>f(\boldsymbol{a}) = L.</math> [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:46, 25 November 2020 (UTC)
::{{ping|D.Lazard}} in my experience this is not the definition of limit commonly used: almost always one takes the hypothesis <math>0 < d(\boldsymbol{x}, \boldsymbol{a})< \delta</math>, specifically removing '''a''' from the ball and allowing the possibility of removable discontinuities. (The article [[limit of a function]] agrees that this is the more common usage.) --[[User:JayBeeEll|JBL]] ([[User_talk:JayBeeEll|talk]]) 16:05, 25 November 2020 (UTC)
:::I have never seen before the definition with <math>0 < d(\boldsymbol{x}, \boldsymbol{a})< \delta,</math> but I must aknowledge that it is used in English Wikipedia. However, in French Wikipedia <math>\boldsymbol{x} = \boldsymbol{a}</math> is not excluded. In German Wikipedia, the Engish definition is given first and is called the "punctured definition"; the French definition is given later in the article, it is presented as "newer" and called "unpunctured". I would guess that the definition used in English Wikipedia is commonly used in US pedagogical mathematics, while the other definition is more commonly used in advanced mathematics. This is only a guess, and would require a verification. As several articles are concerned by this, I'll open a discussion at [[WT:WPM]]. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 17:52, 25 November 2020 (UTC)
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