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If I only know, that the second derivative f''(x) of f at some point x exists (but not necessarily anywhere else) and nothing more (except what follows directly from that, or is implied such as that the first derivative exists in a neighbourhood of x), does it follow that the second derivative at x is represented by the second central difference quotient, that is, does f''(x) = \lim_{h \to 0} \frac{f(x+2h) - 2 f(x) + f(x-2h)}{4h^2} $ = \lim_{h \to 0} \frac{\frac{f(x+2h) - f(x)}{2h} - \frac{f(x) - f(x-2h)}{2h}}{2h} = \lim_{h \to 0} \frac{\frac{f(x+h+h) - f(x+h-h)}{2h} - \frac{f(x-h+h) - f(x-h-h)}{2h}}{2h} $ hold? (Where I added the last two reformulations only as an illustration of how one would arrive at the first expression, namely by changing the direction of the approximation in the definition of f''(x).)
The Wikipedia article on the second derivative states that this should be so, but doesn't give proof or citation (also the premises aren't stated explicitly there) and the Wikipedia article on the characteristic function of a probability distribution cites Lukacs (1970), Corollary 1 to Theorem 2.3.1, in which he seems to say that the above identity holds - assuming that this is clear to the reader. I started thinking about this when reading a text from my university also about characteristic functions which also assumes the above identity is clear to the reader but doesn't seem to give stronger premises either. I think this becomes simple if one assumes a little more, like existence (and continuity) of the second derivative in a neighbourhood, but I would like to know if it holds also in this case with minimal assumptions.
I have been thinking about this for quite a while and my feeling is that a counterexample would have to be quite contrived but I also can't convince myself, that the above should hold always.
Also, although I suspect that these will work in a quite similar manner, what is the situation for higher order derivatives existing at a single point and the central difference approximation? What about noncentral approximations.

Thanks to anyone who can shed any light on this, this has been bugging me for quite a while as I keep thinking that maybe the answer is really simple and I should have been able to figure it out myself a long time ago...

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Assuming that I am understanding your question correctly, you can evaluate that limit by L'Hopital's rule and show that the desired limit is the second derivative.

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    yes, that's what I was look$i$ng for... somehow I didn't think of that... thanks!2012-01-20