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The current text claims that there is no non-visual definition of Local Linearity, but my calc text (Varberg, Purcell, and Rigdon) has a more rigorous definition. I'm hesitant to edit the page, because I am just reaching this section in my studies (not very knowledgeable in the subject), and I don't know how to code math symbols for wiki pages. I just thought it was interesting and probably incorrect that the page claims "there is no other definition".

Local linearity and differentiability of functions are NOT QUITE THE SAME. If a function has a vertical tangent line at a point it is locally linear there but is not differentiable there because the slope of the tangent line is not defined. Otherwise the concepts are the same.

Merge

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It should be considered to merge this page with tangent line since local linearity is precisely the idea being captured by a tangent line. —Preceding unsigned comment added by 98.31.62.207 (talk) 16:41, 4 February 2009 (UTC)[reply]

Not. Lorem Ip (talk) 18:58, 8 September 2010 (UTC)[reply]

Quick Edits to Correct Inaccuracies

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I made some edits, mostly to remove the claim that a locally linear function is smooth-- that is absolutely wrong. I think that the article should either provide a precise definition of local linearity (saying a function has a tangent line is not good enough for me-- does that just mean it's differentiable? that there exists a linear approximation of the function good to second order?), or we should treat local linearity as a non-rigorous idea that gives some intuition of what differentiability means. (I prefer the second option.) I also wonder if points with vertical tangents should be considered locally linear; it seems cleaner to disallow that to me. (In particular, the current list of failures of local linearity excludes those points.) How do we handle ?, ? Low-level explanations should remain, since this term seems to show up primarily in freshman calculus texts, but the article could still use some work, in my opinion.140.114.81.55 (talk) 03:45, 22 October 2010 (UTC)[reply]

This seems hard to "correct" without editorializing

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If this phrase shows up in freshman calculus texts, no wonder students are often confused. The "best option" seems to be to explain that it's bad terminology: how can a function be "locally linear" (at the origin, say) without being linear everywhere?

The "exceptions" merely confuse the issue — it's something of an understatement to claim a nowhere-differentiable continuous function isn't "locally linear" because it has "cusps"!

Moreover, the graph of — assuming we take the real cube root — is continuously differentiable — , even — everywhere. Proof: the map

   

embeds the real line as its graph in ; the coordinates are polynomials and thus trivially analytic. The map isn't differentiable at the origin because "infinity" isn't a linear map.

My vote would be to redirect to "tangent," since editorial policy most likely precludes "calling a spade a spade," and discontinuous derivatives are more a matter of definition than geometry. —Preceding unsigned comment added by 75.184.118.88 (talk) 13:09, 21 December 2010 (UTC)[reply]