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.