In 1942 A. Sard [3] (see also [4]) proved the following theorem.



**Theorem (Sard).** *Let $f:M^m\to N^n$ be of class $C^k$, and let $S={\rm crit}\, f$. 
If $k> \max (m-n, 0)$, then $\mathcal{H}^n(f(S))=0$.*


Here $\mathcal H^s$ stands for the $s$-dimensional
Hausdorff measure (we shall follow the convention that
$\mathcal H^s$ is the counting measure for all $s\leq 0$) and, for a
$C^1$ mapping $f\colon M^m\to N^n$,
${\rm crit}\, f$ denoted the set of critical points of $f$.



It is well known that the assumptions of Sard's theorem
are optimal within the scale of $C^k$ spaces. 
Now, several years after Sard's paper, A.Ya.Dubovitskii [1]
obtained a more general, better result.

**Theorem (Dubovitskii).** *Let $f\colon M^m\to N^n$ be a mapping of class
$C^k$. Set $s=m-n-k+1$. Then
$$
\mathcal H^s (f^{-1}(y) \cap {\rm crit}\, f) = 0 
\quad
\text{for $\mathcal H^n$ a.e. $y\in N^n$.}
$$*

If $k> \max (m-n, 0)$, Sard's theorem follows from that of Dubovitskii. 

Dubovitskii, like a large number of mathematicians in the Soviet
Union of that time, was isolated from the West and from the new
results of western mathematics. He does not quote Sard's paper. On
pages 398-402 of [1] he gives a variant of Whitney's
example for the sharpness of the Sard theorem, and an example of a function $f\in C^k\bigl((0,1)^m,(0,1)^n\bigr)$ such that all sets $f^{-1}(y)\cap
{\rm crit}\, f$ have $(m-n-k)$-dimensional Hausdorff measure greater than
zero, where $m,k,n$ are positive integers such that $m-n-k>0$. He
attributes the first example to Menshov but gives no reference,
and acknowledges Menshov, Novikov, Kronrod and Landis in his
Introduction.


The result of Dubovitskii remained unknown until 2005
when a new proof and some generalizations were published in [2]. It is very surprising since his paper was quoted in Milonor's, *Topology from the Differentiable Viewpoint*
(p. 10).

[1] A. Ya. Dubovickii, On the structure of level sets of
differentiable mappings of an $n$-dimensional cube into a
$k$-dimensional cube. (Russian) Izv. Akad. Nauk SSSR. Ser.
Mat. 21 (1957), 371-408.

[2] B. Bojarski, P. Hajłasz, P., P. Strzelecki, Sard's theorem for mappings in Hölder and Sobolev spaces. Manuscripta Math. 118 (2005), no. 3, 383–397.

[3] A. Sard, The measure of the critical values of
differentiable maps. Bull. Amer. Math. Soc. 48
(1942), 883-890.

[4] S. Sternberg, *Lectures On Differential Geometry.*
Prentice Hall, 1964.

**Personal comment.** I proved the result of Dubovitskii as an undergraduate student. When I discovered that it had already been published, I was quite devastated. I had waited 15 years before I decided to publish it. I am happy I did publish it. Not because of a new `modern' proof and some generalizations that I and my collaborators were able to obtain, but because the old result of Dubovitskii has been brought to public and gained a proper recognition. This comment is related to an <a href="https://mathoverflow.net/a/294969/121665">answer</a> that I gave to another post.