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What s/he is saying, if I understand correctly, is that two and only two kinds of discontinuities in derivatives (and second- and higher-order derivatives) can
by qubex 8y ago
What s/he is saying, if I understand correctly, is that two and only two kinds of discontinuities in derivatives (and second- and higher-order derivatives) can exist: jump derivatives and those exemplified by the pathology inherent in sin(1/x) at the origin thereof; given that spacetime continues beyond the Cauchy horizon but the derivative does not, the first kind (the abrupt ”vertical step” kind) must be ruled out, leaving only the second variety as a candidate. Graph that function, and it’s first and second derivatives around the origin, and you will begin to realise how bizarre this behaviour is and what strange implications it may have. (Then again, this is deep within a black hole, in an area where time and the radial dimension of space are switched, so it’s just a new kind of devilry in an already fraught area.)
- jkabrg 8y agoNo. The relevant theorem is called Darboux's Theorem. It states that if f(x) has a derivative f'(x) everywhere, then f'(x) satisfies the intermediate value property, meaning: If f'(A) < 0 and f'(B) > 0 then there exists a C between A and B such that f'(C) = 0. Why this is interesting is because f'(x) can be a discontinuous function. f'(x) being discontinuous is not the same thing as f(x) not having a derivative at some point; for example, the absolute value function |x| is simply not differentiable at x=0. The following function IS differentiable at x=0 but its derivative at x=0 is discontinuous: f(x) = x^2 sin(1/x) if x != 0 else 0 You can verify that this function satisfies Darboux's theorem. Darboux's theorem implies in particular that if f'(x) exists at some point x and is discontinuous at that x then the discontinuity is not a jump discontinuity. Link to Darboux's theorem: https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis) https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis)