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: Thank both of you for helping to clarify this point. It seems to me that the expression {{math|1/{{var|f}}}} is not a function ''per se'', but might be if one specified its ___domain, codomain, and arguments correctly. Could someone who has a clearer picture of what this is trying to say rewrite it to actually be ''correct'' (i.e. not abuse language by talking about "functions", but rather "expressions")? Or maybe this is totally clear to most readers and I'm just pedantic :) [[User:Siddharthist|siddharthist]] ([[User talk:Siddharthist|talk]]) 05:47, 24 November 2017 (UTC)
::I may have been too sketchy in my preceding post. In any case, this is not a problem of expressions, this is purely a problem of functions: If {{math|''f''}} is a non-constant continuous function (in the sense of this article, that is, its ___domain contains an open set), then, this is a theorem that <math>\boldsymbol x \to 1/f(\boldsymbol x)</math> is also a function whose ___domain may be difficult to specify, even if {{math|''f''}} and its ___domain are well defined. I have edited the article section for clarifying things (I hope). [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 11:06, 24 November 2017 (UTC)
I'll remove the old example of the section for the following reasons. This example is confusing, as it is unclear whether the difficult problem is to find some ball or the largest ball. In the latter case, the example duplicates essentially the new example (reciprocal function), while being more [[WP:TECHNICAL]]. If the problem is to find some ball, it is generally not difficult, at least if one knows an upper bound of the first derivatives in a neighborhood of the center of the ball, which is the case when {{mvar|f}} is a polynomial. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 14:57, 24 November 2017 (UTC)
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