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The Lanczos generalized derivative comes from local regression, and is described in this pdf.
(You do not need to read that to understand my questions.)
If you do read it, observe that I'm pulling out the constant $\frac32$ since that doesn't affect my questions.



Does there exist a continuous function $\: f : \mathbf{R} \to \mathbf{R} \:$ such that for all real numbers $x$,


($\hspace{.01 in}$i) $\quad$ $\displaystyle\lim_{h\to 0}\;\left(\frac1{h^3} \cdot \left(\displaystyle\int_{-h}^h\:\left(t\cdot f(x+t)\right)\:dt\right)\right) \;\;$ does not exist?

($\hspace{.01 in}$ii) $\quad$ if $\;\; \displaystyle\lim_{h\to 0}\;\left(\frac1{h^3} \cdot \left(\displaystyle\int_{-h}^h\:\left(t\cdot f(x+t)\right)\:dt\right)\right) \;\;$ exists then it is infinite?



Whenever $f$ is symmetric about $x$, the limit equals zero (since the integral will be zero independent of $h\hspace{.01 in}$.) $\:$ This rules out all of the continuous nowhere differentiable functions that I am aware of.

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Have you tried any of the standard continuous nowhere differentiable functions? What happens? Also, it might help if you gave some background or references on this notion of derivative. –  Nate Eldredge May 2 '12 at 12:52
    
Have you tried the Baire category approach? –  user31373 Jul 1 '12 at 18:38
    
This construction creates nonsymmetric zigzags: starting with $f(x)=x$, we get a piecewise linear function with slopes $9/4,-3/2,9/4$, etc. I think this might be an example of (i). –  user31373 Jul 28 '12 at 23:39
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