2 fixed critical point/value confusion

Consider a compact sub-manifold $X \subset \mathbb{R}^n$ of Euclidean space and let $f:X \to \mathbb{R}$ be any smooth function. Recall that $c x \in \mathbb{R}$ X$is a critical value point of$f$if the gradient$\nabla_c \nabla_x f$is identically zero. , and in this case$f(x) \in \mathbb{R}$is called a critical value of$f$. It is well-known by Sard's theorem that the set of critical values of$f$has Lebesgue measure zero as a subset of$\mathbb{R}$. Is there a "fuzzy" version of this theorem for a suitable class of smooth functions? More precisely, define the monotone function$M_f : \mathbb{R}^+ \to \mathbb{R}^+$as follows:$M_f(\epsilon)$is the Lebesgue measure of the set$\lbrace $\lbrace p \in \mathbb{R} \mid p = f(x) \|\nabla_p text{ for some } x \in X \text{ with }\|\nabla_x f \| < \epsilon\rbrace$. epsilon\rbrace. By Sard's theorem, $M_f(0) = 0$ for any smooth $f$.

Here's the question:

Can we classify those smooth $f:X \to \mathbb{R}$ for which $M_f$ is continuous on its domain of definition in general (and in particular, continuous from above at $0$)?

Essentially, we know that the measure of values of $f$ where the gradient of the pre-image equals zero is zero. When the gradient comes within $\epsilon$ of zero, can we bound the measure by a continuous function of $\epsilon$?

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# Measuring almost-critical values of smooth functions.

Consider a compact sub-manifold $X \subset \mathbb{R}^n$ of Euclidean space and let $f:X \to \mathbb{R}$ be any smooth function. Recall that $c \in \mathbb{R}$ is a critical value of $f$ if the gradient $\nabla_c f$ is identically zero. It is well-known by Sard's theorem that the set of critical values of $f$ has Lebesgue measure zero as a subset of $\mathbb{R}$.

Is there a "fuzzy" version of this theorem for a suitable class of smooth functions?

More precisely, define the monotone function $M_f : \mathbb{R}^+ \to \mathbb{R}^+$ as follows: $M_f(\epsilon)$ is the Lebesgue measure of the set $\lbrace p \in \mathbb{R} \mid \|\nabla_p f \| < \epsilon\rbrace$. By Sard's theorem, $M_f(0) = 0$ for any smooth $f$.

Here's the question:

Can we classify those smooth $f:X \to \mathbb{R}$ for which $M_f$ is continuous on its domain of definition in general (and in particular, continuous from above at $0$)?

Essentially, we know that the measure of values of $f$ where the gradient equals zero is zero. When the gradient comes within $\epsilon$ of zero, can we bound the measure by a function of $\epsilon$?