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Is it true always that $f'(x)$ whenever is discontinuous at a point , that implies f(x) being non differentiable at that point ? Or is the vice ? As for example $f(x) = max(-x,x)$ derivative is discontinuous at point $x= 0$ and so $f(x)$ is also non differentiable at $x= 0$ . By discontinuous i mean function limit doesnt exist there .

Paracetamol
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  • Consider $$f(x)=x^2\cdot(\text{some weird but bounded function defined on a neighbourhood of $0$})$$ at $x=0$. – Michael Hoppe May 10 '22 at 10:15
  • A function can be differentiable everywhere, but have a derivative that is "very" discontinuous https://math.stackexchange.com/questions/292275/discontinuous-derivative – Joe May 10 '22 at 10:40
  • But at x= 0 its actually continuous isnt ? We just say limit doesnt exist because of the fact that its going to infinity but if we try plotting it , it intuitvely feels like it should be continuous right ? @Joe – Paracetamol May 10 '22 at 12:55
  • @MichaelHoppe can you give the example for your function except the general form type one given by Joe ? – Paracetamol May 10 '22 at 12:57
  • Try $x^2\sin(1/x)$. – Michael Hoppe May 10 '22 at 14:51

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