= Solution
First establish the limiting law needed by the <delta method>. For fixed $t$, put $u_n=1/2+t/\sqrt n$ and let $N_n$ count how many of the $U_i$ are at most $u_n$. Then $N_n$ has a <binomial distribution> with <expectation> $n/2+t\sqrt n$ and <variance> $n/4-t^2$. The event $M_U\leq u_n$ is exactly $N_n\geq(n+1)/2$. Its standardized threshold tends to $-2t$, so the <central limit theorem> for these binomial counts gives
$$
\Pr\!\left(\sqrt n(M_U-1/2)\leq t\right)\longrightarrow\Phi(2t).
$$
Here $\Phi$ is the <cumulative distribution function> of the <standard normal distribution>. Equivalently,
$$
\sqrt n(M_U-1/2)\xrightarrow{d}N(0,1/4).
$$
Use the usual local regularity for the <delta method>: $f$ is continuous at the population <median> $\theta$ and $f(\theta)>0$. The <inverse function theorem>, applied at $F(\theta)=1/2$, gives
$$
\left.\frac{d}{du}F^{-1}(u)\right|_{u=1/2}=\frac1{f(\theta)}.
$$
Therefore
$$
\boxed{\sqrt n(M_X-\theta)\xrightarrow{d}
N\!\left(0,\frac1{4f(\theta)^2}\right).}
$$
The <asymptotic normality> gives the requested leading <variance> approximation
$$
\boxed{\operatorname{avar}(M_X)=\frac1{4n f(\theta)^2}.}
$$
Here the variance parameter for the rescaled limiting law is $1/(4f(\theta)^2)$, while the source states the corresponding $1/n$ approximation for the unscaled <estimator>. Strict increase of $F$ alone does not force $f(\theta)>0$; that local condition is needed for this form of the <asymptotic distribution of a sample median>.
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