Talk:Local linearity

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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.

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

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 ${\displaystyle {\sqrt[{3}]{x}}}$?, ${\displaystyle |{\sqrt[{3}]{x}}|}$? 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

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 ${\displaystyle x\mapsto {\sqrt[{3}]{x}}}$ — assuming we take the real cube root — is continuously differentiable — ${\displaystyle C^{\omega }}$, even — everywhere. Proof: the map

   ${\displaystyle c(t)={\binom {t^{3}}{t}}}$

embeds the real line as its graph in ${\displaystyle \mathbb {R} ^{2}}$; 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