On the modulus of continuity of functions whose image has positive measure, and metric embeddings into $\mathbb{R}^d$ without shrinking

A generalization of the classical Sard theorem in the plane is the following. Let $f$ be a function defined on a subset $A\subset{\mathbb R}^2$. If $f$ has modulus of continuity $\omega(r)\lesssim r^2$, then $f(A)\subset{\mathbb R}$ has Lebesgue measure zero. Choquet claimed in \cite{Choquet} that this was a full characterization, i.e. for every $\omega$ for which $\omega(r)/r^2$ converges to $\infty$ as $r\to 0$, there is a counterexample. We disprove this by showing that the correct characterization, in $\mathbb{R}^d$, is $\int_{0}^{1} \omega(r)^{-1/d}=\infty$. For the precise statement see Theorem 2. We obtain this as a special case of a more general result. We study which spaces $(X,\rho)$ can be embedded into ${\mathbb R}^d$ without decreasing any of the distances in $X$. That is, we ask the question whether there is an $f: X\to {\mathbb R}^d$ such that $\|f(x)-f(y)\|\ge \rho(x,y)$ for every $x,y\in X$. We study this problem for some very general distance functions $\rho$ (we do not even assume that it is a metric space, in particular, we do not assume that $\rho$ satisfies the triangle inequality), and find quantitative necessary and sufficient conditions under which such a mapping exists. We will obtain the characterization mentioned above as a special case of our metric embedding results, by choosing $X$ to be an interval in $\mathbb{R}$, and defining $\rho$ by putting $\rho(x,y)=r$ if $\|x-y\|=\omega(r)$.