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The backward heat equation $u_t=-D{u}_{xx}$, $D>0$, amplifies a spatial Fourier mode by $e^{D{k}^{2}t}$. On a periodic domain, fix $T>0$ and take initial data $f_n=e^{-D{n}^{2}T/2}\sin (nx)$. Their L2 norms tend to zero, while the corresponding solutions at $T$ have L2 norms growing without bound. Thus the solution map is not continuous in the ordinary L2 norm; this is an ill-posed problem.^[ref]

Real analyticity alone does not resolve the issue. The periodic analytic function $f(x)={\sum }_{n\ge 1}{e}^{-n}\cos (nx)$ has exponentially decaying coefficients, but after any positive time the proposed coefficients $e^{D{n}^{2}t-n}$ are not square summable. The required data restriction must control growth on the Gaussian frequency scale. For example, a fixed-horizon normgives $\Vert u(T){\Vert }_2=\Vert f{\Vert }_{X_T}$ when the Fourier basis is normalized. This very strong data norm changes the stability question. In inverse heat reconstruction, an explicit regularization or frequency cutoff is needed when measured data carry ordinary small-scale noise.

An inverse problem is an ill-posed problem if it fails any requirement of a well-posed problem: existence for every admissible datum, uniqueness, or continuity of the reconstructed object as the data change. These requirements depend on the chosen spaces and norms, not just the equation. Clason's regularization notes formulate this distinction explicitly.^[ref]

Complete averaging illustrates lost information:The difference mode $q_{-}=(1,-1{)}^T/\sqrt{2}$ belongs to the null space because $B{q}_{-}=0$. Even restricting both pixels to $[0,1]$ leaves infinitely many indistinguishable inputs. This is failure of uniqueness. Data such as $(0.51,0.49{)}^T$ also have no exact preimage under $B$, illustrating failure of existence when arbitrary two-component observations are admissible.

The two-pixel matrix $A$ in the inverse problem example has a weakly observed difference mode, not a null space mode. Its inverse is continuous: this finite-dimensional problem is a well-posed problem with noise-sensitive conditioning. Small positive singular values alone do not imply mathematical ill-posedness. In infinite-dimensional problems, positive singular values approaching zero can make inversion unbounded: unit solution modes then produce data tending to zero. This destroys continuity even without a null space. Regularization of an inverse problem must distinguish missing information from information recoverable only at the cost of amplifying measurement noise.

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 $h$.