Does local Hölder-continuity suffice to get an upper bound for Hausdorff dimension - or does it need to be global? ((Mistake in the literature?)) It is a well-known fact that Brownian Motion is almost surely locally $\alpha$-Hölder continuous for $\alpha\in(0,1/2)$. This is sometimes used in conjunction with the following result concerning Hausdorff dimension -- when proving McKean's Theorem, for instance (that is, Proposition 2 strengthened to equality).
Proposition 1. Let $(M,d_1)$ and $(N,d_2)$ be two metric spaces, and $f:M\to N$ a (globally) $\gamma$-Hölder continuous function with $\gamma$-Hölder constant $L$. Then we have, for all $\alpha\geq0$, and for all subsets $A$ of $M$, that
\begin{equation*}
    \mathcal{H}^{\infty}_{\alpha/\gamma}(f(A))\leq L^{\alpha/\gamma}\cdot\mathcal{H}^{\infty}_{\alpha}(A)
\end{equation*}
In particular, we have that $\dim f(A)\leq\frac{1}{\gamma}\dim A$. $\Diamond$
Here, we denote by $\mathcal{H}^{\infty}_{\alpha}(A)$ the unlimited $\alpha$-Hausdorff content of $A$. One then uses Proposition 1, in some way, to show that
Proposition 2. Let $\{B(t):t\geq0\}$ be $d$-dimensional Brownian motion, and fix a subset $A$ of $[0,\infty)$. Then, almost surely,
$\dim B(A)\leq\min(2\dim A,d)$. $\Diamond$
since we already know that Brownian motion is almost surely locally $\alpha$-Hölder continuous. My question is: How can we bridge the gap from Hölder continuity in Proposition 1 to just local Hölder continuity?
All books I have taken a look at either ignore the distinction between the two -- or just remark that it is sufficient if $f$ is only locally Hölder continuous, without providing a proof. This is, for example, done in Brownian Motion (2010) by Peter Mörters and Yuval Peres, with Remark 4.15 on page 102 (here is a publicly available pdf, provided by the University of Bath). It seems that the three Bachelor's thesis/essays here, here and here have relied upon that remark without further justification. On the other hand, the books treating Hausdorff measure by Falconer (2003) and Mattila (1995) only state results for globally Hölder continuous functions, and make no mention of strengthened versions for local Hölder continuity.
So far, I have tried the following to prove Proposition 1 for locally Hölder continuous functions, but haven't made much progress.
Proposition 3. If $(M,d_1)$ is a separable metric space, we can replace the condition "$f$ is $\gamma$-Hölder continuous" in Proposition 1 with "$f$ is locally $\gamma$-Hölder continuous everywhere". $\Diamond$
Attempt of proof. Since $M$ is separable, there exists a countable, dense subset $X$ of $M$. Then since $f$ is locally $\gamma$-Hölder continuous everywhere, there exist, for each point $x$ in $X$, two constants $\delta_{x}>0$ and $L_{x}\geq0$ so that for all $y\in M$ with $d_1(x,y)<\delta_x$, we have $d_2(f(x),f(y))\leq L_x\cdot\left(d_1(x,y)\right)^{\gamma}$. As $X$ is dense in $M$, we have
\begin{equation*}
    f(A)=\bigcup_{x\in X}f(B(x,\delta_x))
\end{equation*}
and so, by countable stability of $\mathcal{H}^{\infty}_{\alpha}$ and by Proposition 1, it is sufficient to show that $f\vert_{B(x,\delta_x)}$ is $\gamma$-Hölder continuous for all $x\in X$. Indeed, fix $x\in X$ and let $y,z$ be two points in $B(x,\delta_x)$. We see that
\begin{equation*}
    d_2(f(x),f(y))\leq L_{x}\cdot\left(d_1(x,y)\right)^{\gamma},\quad d_2(f(x),f(z))\leq L_{x}\cdot\left(d_1(x,z)\right)^{\gamma}
\end{equation*}
and, therefore, that
\begin{equation*}
    d_2(f(y),f(z))\leq L_{x}\cdot\left(d_1(x,y)\right)^{\gamma}+L_{x}\cdot\left(d_1(x,z)\right)^{\gamma}
\end{equation*}
How to go on? $\Diamond$
The definitions of Hölder continuity and local Hölder continuity I'm am referring to here are as below.
Definition. Let $(M,d_1)$ and $(N,d_2)$ be two metric spaces, $f:M\to N$ a function and $\gamma>0$ a constant. Then we say that $f$ is $\gamma$-Hölder continuous if there exists a constant $L\geq0$ so that, for all $x,y\in M$,
\begin{equation*}
    d_2(f(x),f(y))\leq L\cdot\left(d_1(x,y)\right)^{\gamma}
\end{equation*}
We then also say that $f$ has $\gamma$-Hölder constant $L$. $\Diamond$
Definition. Let $(M,d_1)$ and $(N,d_2)$ be two metric spaces, $f:M\to N$ a function, $\gamma>0$ a constant and $x\in M$ a point. Then we say that $f$ is locally $\gamma$-Hölder continuous at $x$ if there exist constants $\delta>0$ and $L\geq0$ so that, for all $y\in M$ with $d_1(x,y)<\delta$,
\begin{equation*}
    d_2(f(x),f(y))\leq L\cdot\left(d_1(x,y)\right)^{\gamma}
\end{equation*}
We then also say that $f$ has local $\gamma$-Hölder constant $L$ at $x$. $\Diamond$
Definition. Let $(M,d)$ be a metric space and $E$ a subset of $M$. For $\alpha\geq0$, we define the unlimited $\alpha$-Hausdorff content of $E$ by
\begin{equation*}
\mathcal{H}_{\alpha}^{\infty}(E):=\inf\left\{\sum_n\left\vert F_n\right\vert^{\alpha}:\{F_n\}_n\text{ is a cover of }E\right\}
\end{equation*}
Further, we define the Hausdorff dimension of $E$ by
\begin{equation*}
\dim E:=\inf\{\alpha\in[0,\infty):\mathcal{H}_{\alpha}^{\infty}(E)=0\}
\end{equation*}
Note that both $\mathcal{H}_{\alpha}^{\infty}(E)$ and $\dim E$ are possibly infinite. $\Diamond$
 A: The issue is the definition of local Holder continuity.  In general, we say that a property $P$ of metric spaces is satisfied locally in a metric space $M$, if every point in $M$ has an open neighborhood that satisfies $P$.
Specifically, The correct definition is:
Definition. Let $(M,d_1)$ and $(N,d_2)$ be two metric spaces, $f:M\to N$ a function, $\gamma>0$ a constant.  We say that $f$ is locally $\gamma$-Hölder continuous in $M$ if for every $x \in M$, there exist     $\delta, L>0$ so that, for all $y,z$ that satisfy $d_1(y,x)<\delta$ and $d_1(z,x)<\delta$,  we have
\begin{equation*}
    d_2(f(z),f(y))\leq L\cdot\left(d_1(z,y)\right)^{\gamma}
\end{equation*}
$\Diamond$
See https://second.wiki/wiki/lokale_hc3b6lderstetigkeit
In particular, if $f$ satisfies the definition above, then
$(*)$  The restriction of $f$ to any compact subset $K$ of $M$ is $\gamma$-Holder.
The  requirement $(*)$ is equivalent to the definition above if $M$ is locally compact.
The   requirement $(*)$ is the definition of local Holder continuity used in line 7 of
https://en.wikipedia.org/wiki/H%C3%B6lder_condition#H%C3%B6lder_spaces
Once one uses the above definition, the required inequality for Hausdorff dimension follows easily from the same inequality for globally Holder functions, separability of $M$ and countable stability of Hausdorff dimension.
