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Given a function, is it always standard to determine its differentiability along the two lines $y=0$ and $y=x$ in order to check whether it is differentiable or not? Are there any other lines that can be used?

Artemisia
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  • There are many other lines one can check! Differentiability requires that it be differentiable along any differentiable path leading to the point in question. – Dustan Levenstein Dec 04 '13 at 12:41
  • But we can't really check all lines, can we? – Artemisia Dec 04 '13 at 12:43
  • Not typically; there are other techniques for determining differentiability. – Dustan Levenstein Dec 04 '13 at 12:45
  • Like checking whether partial derivatives exist and are equal? And continuity? – Artemisia Dec 04 '13 at 12:49
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    That is one simple technique. The most general method is to look for the Jacobian matrix (using partial derivatives), and then verify that it satisfies the constraint shown here. – Dustan Levenstein Dec 04 '13 at 12:52
  • However, usually, I have seen the approach along a line... I have never come across the Jacobian in this context. – Artemisia Dec 04 '13 at 12:55
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    That's probably because no one expects you to be able to verify differentiability in this way, since it is harder to understand/apply. – Dustan Levenstein Dec 04 '13 at 12:58
  • Ah. But as a standard, will the method of directional derivatives or along lines work? – Artemisia Dec 04 '13 at 12:59
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    If you mean just straight lines, then no. The same link I gave you also gives an example of a function which has all directional derivatives (i.e., derivatives along straight lines), but is not differentiable, because it fails to have a derivative along a different type of curve. – Dustan Levenstein Dec 04 '13 at 13:01
  • you can consider even more complicated paths! Actually, all possible paths have to considered to prove differentiability (like in continuity) – Avitus Dec 04 '13 at 13:16
  • A general trend that I have observed... functions with a maximum degree of 2 usually work when I use the test along $y=0$ and $y=x$. Degree 3 can be checked along a curve $y=x^2$. – Artemisia Dec 04 '13 at 13:21
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    @Artemisia: This related problem may help your intuition. – Mark Fantini Dec 07 '13 at 20:14

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