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I've been Studying Differentiability theorems

Note: => means comply

!=> mean doesn't comply

  1. Differentiability => Continuity
  2. Continuity !=> Differentiability
  3. Non Differentiability !=> DisContinuity.
  4. Discontinuity !=> Non Differentiability

I don't understand the last theorem how Discontinuous fn can be Differentiable. Can any tell me why is that cause I new to calculus.. Thanks in advance..

anomaly
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Anwar Aftab
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  • (4) is just the contrapositive of (1), so it does in fact hold. – anomaly Oct 10 '22 at 01:35
  • For single variable functions, it's not possible, see this qs for a dup question. For multivariable functions, it seems that it is possible, see here. – ketsi Oct 10 '22 at 01:37
  • @anomaly . Should 4. not be Discontinuity$\Rightarrow$ Non Differentiablity ? The $!$ seems wrong. – Kurt G. Oct 10 '22 at 10:58
  • @KurtG.: Yeah, that's what I'm referring to. – anomaly Oct 10 '22 at 11:57
  • @GarethMa: I'm not sure what domain the OP is working with, but it's possible for a function $f:\mathbb{R}^n \to \mathbb{R}$ to have all partial derivatives $\partial f / \partial x_j$ exist on a neighborhood of a point $p\in \mathbb{R}^n$ but $f$ not be differentiable on a neighborhood of $p$; that is, it may not satisfy $f(x) = f(p) + L(x - p) + o(|x - p|)$ as $x\to p$ for some linear map $L$. But if the latter does hold, then $f$ must be continuous near $p$; we have $|f(x) - f(p)| = |L| O(|x - p|)$ as $x\to p$. I'm not sure what flavor of differentiability the OP is referring to. – anomaly Oct 10 '22 at 12:09
  • @anomaly: To be fair, I don't know much about multivariable calculus either, it's just googled results that might help, since most people on Math.SE are pretty bad at googling, I must say. Also, "Differentiability Theorem" seems to be reserved for multivariable multidimensional functions, e.g. here. Not sure though, maybe OP can reply ;) – ketsi Oct 10 '22 at 16:41

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