Approximating Error in $\partial_x u(x;\eta)$

[bruno67]

I have a quantity U(x), x>0, which I cannot calculate exactly. Numerically, I can calculate an approximation u(x;\eta), for \eta>0, which is very close to U(x) if \eta is small enough. I know that the error \xi(x;\eta)=u(x;\eta)-U(x) satisfies an estimate
|\xi(x;\eta)|\le E(x;\eta)
where \lim_{\eta\to 0}E(x;\eta)=0 for all x, and I can use this to choose my parameter \eta so that the error lies under a specified tolerance.

Based only on the above, is it possible to derive an approximate estimate for the error in \partial_x u(x;\eta), i.e. \partial_x \xi(x;\eta)?

 

[fresh_42]

No. As far as I can tell from the description, we have good information for any specific value of $x$ but none about the dependence of $x$. And even if everything tends to zero, there could still be high slopes locally.