Ill-posedness of numerical differentiation ($\Vert f_k{\Vert }_{\infty }\to 0\text{ need not imply }\Vert f_k^{\prime }{\Vert }_{\infty }\to 0$)

Recovering a derivative from values controlled only in the uniform norm is an ill-posed problem: the differentiation map is not continuous in those norms. On $[0,1]$, take $f_k(x)=\sin (kx)/k$. Then $\Vert f_k{\Vert }_{\infty }\le 1/k\to 0$, while $f_k^{\prime }(x)=\cos (kx)$ has uniform norm one for every $k$. Arbitrarily small value perturbations can therefore conceal nonvanishing derivative changes.

Numerical differentiation can still be useful when additional smoothness bounds, a suitable sampling scale, or a regularization method control the admissible changes. The basic noise amplification by differentiation of a linear interpolant gives a finite-data version: endpoint uncertainty of size $\delta$ produces derivative uncertainty of order $\delta /h$ at spacing $h$.

For linear interpolation at zero and $h>0$, an error $\delta$ in the second sampled value, with the first exact, changes the linear interpolant by $\Delta p(x)=\delta x/h$. On $[0,h]$, its uniform norm is exactly $|\delta |$, whereas the derivative change is the constant $\delta /h$. With independent absolute bounds $|{\delta }_0|,|{\delta }_1|\le \delta$ on both samples, the value change remains at most $\delta$ and the derivative change is at most $2\delta /h$, attained for opposite endpoint errors.

Thus bounded absolute sample noise can remain bounded under value interpolation while becoming unbounded under numerical differentiation as $h\to 0$. Fixed nonzero sample noise also prevents exact convergence of the value approximation; stability against perturbations and convergence to exact data are different assertions. For a second-sample error $aϵ(h)$ with fixed $a>0$, value convergence requires $ϵ(h)\to 0$, while derivative convergence requires $ϵ(h)=o(h)$ in little o notation, in addition to vanishing truncation error.