We comment on a problem of Mazur from “The Scottish Book" concerning second partial derivatives. We prove that if a function f(x,y) of real variables defined on a rectangle has continuous derivative with respect to y and for almost all y the function $F_y\left(x\right):=f_y^{\text{'}}(x,y)$ has finite variation, then almost everywhere on the rectangle the partial derivative $f_{yx}^{\text{'}\text{'}}$ exists. We construct a separately twice differentiable function whose partial derivative $f_x^{\text{'}}$ is discontinuous with respect to the second variable on a set of positive...

We show that if n is a positive integer and $2^{\aleph ₀}\le \aleph ₙ$, then for every positive integer m and for every real constant c > 0 there are functions $f₁,...,f_{n+m}:ℝⁿ\to ℝ$ such that $(f₁,...,f_{n+m})\left(ℝⁿ\right)={ℝ}^{n+m}$ and for every x ∈ ℝⁿ there exists a strictly increasing sequence (i₁,...,iₙ) of numbers from 1,...,n+m and a w ∈ ℤⁿ such that $(f_{i₁},...,f_{iₙ})\left(y\right)=y+w$ for $y\in x+(-c,c)\times {ℝ}^{n-1}$.

Let $\Lambda \subset {ℝ}^n\times {ℝ}^m$ and $k$ be a positive integer. Let $f:{ℝ}^n\to {ℝ}^m$ be a locally bounded map such that for each $(\xi ,\eta )\in \Lambda$, the derivatives $D_{\xi }^{j}f\left(x\right):=\frac{d^j}{d{t}^j}f(x+t\xi ){|}_{t=0}$, $j=1,2,\cdots k$, exist and are continuous. In order to conclude that any such map $f$ is necessarily of class $C^k$ it is necessary and sufficient that $\Lambda$ be not contained in the zero-set of a nonzero homogenous polynomial $\Phi (\xi ,\eta )$ which is linear in $\eta =({\eta }_1,{\eta }_2,\cdots ,{\eta }_m)$ and homogeneous of degree $k$ in $\xi =({\xi }_1,{\xi }_2,\cdots ,{\xi }_n)$. This generalizes a result of J. Boman for the case $k=1$. The statement and the proof of a theorem of Boman for the case $k=\infty$ is also extended...

We study possible Borel classes of sets of Fréchet subdifferentiability of continuous functions on reflexive spaces.