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If $f\in C^1(\mathbb{R}^2,\mathbb{R})$, then the set of critical values may have positive measure.

The classical construction due to Whitney was mentioned in the answer by T. Amdeberhan. However, the simplest one is due to E.L.Grinberg. The idea is as follows:

If $C$ is the Cantor middle thirds set, then $C+C=[0,2]$.

There is a $C^1$ function $g:\mathbb{R}\to\mathbb{R}$ such that the critical values of $g$ contain $C$.

Now $f:\mathbb{R}^2\to\mathbb{R}$ defined by $f(x,y)=g(x)+g(y)$ is of class $C^1$ and $C+C=[0,2]$ is contained in the set of critical values of $f$.

If $f\in C^1(\mathbb{R}^2,\mathbb{R})$, then the set of critical values may have positive measure.

The classical construction due to Whitney was mentioned in the answer by T. Amdeberhan. However, the simplest one is due to

E.L.Grinberg, On the smoothness hypothesis in Sard's theorem.  Amer. Math. Monthly 92 (1985), 733–734. 

The idea is as follows:

If $C$ is the Cantor middle thirds set, then $C+C=[0,2]$.

There is a $C^1$ function $g:\mathbb{R}\to\mathbb{R}$ such that the critical values of $g$ contain $C$.

Now $f:\mathbb{R}^2\to\mathbb{R}$ defined by $f(x,y)=g(x)+g(y)$ is of class $C^1$ and $C+C=[0,2]$ is contained in the set of critical values of $f$.

If $f\in C^1(\mathbb{R}^2,\mathbb{R})$, then the set of critical values may have positive measure.

The classical construction due to Whitney was mentioned in the answer by T. Amdeberhan. However, the simplest one is due to E.L.Grinberg. The idea is as follows:

If $C$ is the Cantor middle thirds set, then $C+C=[0,2]$.

There is a $C^1$ function $g:\mathbb{R}\to\mathbb{R}$ such that the critical values of $g$ contain $C$.

Now $f:\mathbb{R}^2\to\mathbb{R}$ defined by $f(x,y)=g(x)+g(y)$ is of class $C^1$ and $C+C=[0,2]$ is contained in the set of critical values of $f$.

If $f\in C^1(\mathbb{R}^2,\mathbb{R})$, then the set of critical values may have positive measure.

The classical construction due to Whitney was mentioned in the answer by T. Amdeberhan. However, the simplest one is due to E.L.Grinberg. The idea is as follows:

If $C$ is the Cantor middle thirds set, then $C+C=[0,2]$.

There is a $C^1$ function $g:\mathbb{R}\to\mathbb{R}$ such that the critical values of $g$ contain $C$.

Now $f:\mathbb{R}^2\to\mathbb{R}$ defined by $f(x,y)=g(x)+g(y)$ is of class $C^1$ and $C+C=[0,2]$ is contained in the set of critical values of $f$.

If $f\in C^1(\mathbb{R}^2,\mathbb{R})$, then the set of critical values may have positive measure. The classical construction due to Whitney was mentioned in the answer by T. Amdeberhan. However, the simplest one is due to E.L.Grinberg. The idea is as follows:

If $C$ is the Cantor middle thirds set, then $C+C=[0,2]$.

There is a $C^1$ function $g:\mathbb{R}\to\mathbb{R}$ such that the critical values of $g$ contain $C$.

Now $f:\mathbb{R}^2\to\mathbb{R}$ defined by $f(x,y)=g(x)+g(y)$ is of class $C^1$ and $C+C=[0,2]$ is contained in the set of critical values of $f$.

If $f\in C^1(\mathbb{R}^2,\mathbb{R})$, then the set of critical values may have positive measure.

The classical construction due to Whitney was mentioned in the answer by T. Amdeberhan. However, the simplest one is due to E.L.Grinberg. The idea is as follows:

If $C$ is the Cantor middle thirds set, then $C+C=[0,2]$.

There is a $C^1$ function $g:\mathbb{R}\to\mathbb{R}$ such that the critical values of $g$ contain $C$.

Now $f:\mathbb{R}^2\to\mathbb{R}$ defined by $f(x,y)=g(x)+g(y)$ is of class $C^1$ and $C+C=[0,2]$ is contained in the set of critical values of $f$.