TITLE: Analogues of Luzin's theorem
QUESTION [25 upvotes]: If $X$ is a compact metric space and $\mu$ is a Borel probability measure on $X$, then the space $C(X)$ of continuous real-valued functions on $X$ is a closed nowhere dense subset of $L^\infty(X,\mu)$, and hence bounded measurable functions are generically discontinuous.  Nevertheless, Luzin's theorem says that every measurable function is in fact continuous on a set of arbitrarily large measure.  This allows us to gain continuity from measurability at the cost of ignoring a small portion of $X$.
Question: Are there any analogues of Luzin's theorem that allow us to go from continuity to Hölder continuity?
A direct analogue would be a statement that given a continuous function $f\in C(X)$ and an arbitrary $\epsilon>0$, there exists a set $X_\epsilon \subset X$ such that $\mu(X_\epsilon) > 1-\epsilon$ and the restriction of $f$ to $X_\epsilon$ is Hölder continuous.  (For my purposes, it would be all right if the Hölder exponent and coefficient become arbitrarily bad as $\epsilon\to 0$.)
Another possible analogue, and one that I am actually more interested in, would be a statement that given a continuous function $f\in C(X)$ and an arbitrary $\epsilon>0$, there exists a set $X_\epsilon \subset X$ such that the restriction of $f$ to $X_\epsilon$ is Hölder continuous (again with arbitrarily bad exponent and coefficient) and instead of an estimate on the measure of $X_\epsilon$, we have
$$
\dim_H(X_\epsilon) > \dim_H(X) - \epsilon,
$$
where $\dim_H$ is Hausdorff dimension.
Full motivation:  Ideally I would like to consider the setting where $T\colon X\to X$ is a continuous map, and obtain a similar statement about the restriction of a continuous potential $f\in C(X)$ to a set of large topological pressure,
$$
P_{X_\epsilon}(f) > P_X(f) - \epsilon,
$$
such that $f$ restricted to $X_\epsilon$ has the Walters property, which deals with variation on Bowen balls rather than on metric balls.  But the purely static version stated above for Hausdorff dimension seems like a good place to start.  Does anybody know of any results in this direction?  Or counterexamples showing that such a theorem can't be true in full generality?
Edit: I've accepted Anonymous's answer, which shows quite nicely that the direct analogue (using measures) fails.  However, I remain very interested in the indirect analogue (using dimensions), which seems to still have a chance of holding, so any information in that direction would be welcomed.

REPLY [2 votes]: I'm posting some references given to me by Jon Chaika as a CW answer, which addresses the question of whether we can get a set $X_\alpha$ such that the restriction to $X_\alpha$ is Hölder continuous.  It seems this question was posed by Márton Elekes in a 2004/5 paper, where a partial answer was given: for every $\alpha\in (0,1]$, a typical continuous function $[0,1]\to\mathbb{R}$ is not $\alpha$-Hölder on any set of Hausdorff dimension greater than $1-\alpha$.  The converse direction was provided in the 2009 PhD thesis of András Máthé, and appeared in a 2013 paper: for every $\alpha$ and every Borel measurable $f\colon [0,1]\to \mathbb{R}$, there is a set $X_\alpha\subset [0,1]$ with Hausdorff dimension $1-\alpha$ such that $f|_{X_\alpha}$ is $\alpha$-Hölder.