Given a measurable subset $P\subset\mathbb R^n$ of possitive $n$-measure the notion of $k$-quasismooth functions $f\colon P\to\mathbb R$ is defined, $k\ge 0$, $k$-integer. This class is characterized in terms of approximate $k$-th order total Peano differentiablity at almost every point of $P$. For $k=0$ we obtain the Egorov–Denjoy–Lusin structure theory of measurable functions. The case $k\ge 1$ connects Lusin's theory with H. Whitney's theory of $k$-smooth functions on arbitrary closed subsets of $\mathbb R^n$. Applications to harmonic analysis, singular integrals, potential theory and PDE will be given.