= Solution
The <empirical distribution function> is
$$
\boxed{F_n(t)=\frac1n\sum_{i=1}^n\mathbf1_{\{X_i\leq t\}},\qquad t\in\mathbb R.}
$$
If $W$ is a standard <Brownian motion> on $[0,1]$, the standard <Brownian bridge> is $\mathbb G(u)=W(u)-uW(1)$. It is a centred <Gaussian process> with continuous paths, endpoints zero, and <covariance>
$$
\mathbb E[\mathbb G(u)\mathbb G(v)]=\min(u,v)-uv.
$$
The <F-Brownian bridge> is the time-changed <Gaussian process> $\mathbb G_F(t)=\mathbb G(F(t))$. Its <covariance> is $F(\min(s,t))-F(s)F(t)$, exactly the covariance of the <empirical distribution process> $\alpha_n=\sqrt n(F_n-F)$.
The <Donsker theorem for empirical distribution functions> states that for every <cumulative distribution function> $F$,
$$
\boxed{\sqrt n(F_n-F)\ \rightsquigarrow\ \mathbb G_F\quad\text{in }\ell^\infty(\mathbb R),\text{ with the supremum norm}.}
$$
Thus this is a function-space <central limit theorem>, not merely convergence of each finite collection of evaluations. The limit is a tight centred <Gaussian process>. No continuity of $F$ is required for this statement; a jump of $F$ can give a jump of $\mathbb G_F$.
For precision, $\ell^\infty(\mathbb R)$ is generally nonseparable, and the finite-sample process need not be a Borel-measurable random element for its norm topology. The displayed <convergence in distribution> has the usual empirical-process interpretation: for every bounded continuous real functional $H$, the <outer expectation> and inner expectation of $H(\alpha_n)$ converge to $\mathbb EH(\mathbb G_F)$. The <outer expectation> is the infimum of expectations of measurable majorants, and the inner expectation is its negative applied to $-H(\alpha_n)$. For measurable real statistics this gives ordinary <convergence in distribution>. The limit is tight and separably supported: $g\mapsto g\circ F$ maps the separable space of continuous bridge paths continuously into $\ell^\infty(\mathbb R)$.
The map $z\mapsto\|z\|_\infty$ is Lipschitz, since
$$
\bigl|\|z\|_\infty-\|w\|_\infty\bigr|\leq\|z-w\|_\infty.
$$
Consequently the <continuous mapping theorem> applied to the <Donsker theorem for empirical distribution functions> gives $\|\alpha_n\|_\infty\xrightarrow d\|\mathbb G_F\|_\infty$. The statistic on the left is measurable: both $F_n$ and $F$ are right-continuous, so its supremum equals the supremum over rational $t$.
If $F$ is continuous, its limits at $-\infty$ and $+\infty$ are zero and one. The <intermediate value theorem> therefore shows that $F(\mathbb R)$ contains $(0,1)$, even when $F$ is not strictly increasing. A continuous <Brownian bridge> is zero at its endpoints, so pathwise
$$
\sup_{t\in\mathbb R}|\mathbb G(F(t))|=\sup_{0<u<1}|\mathbb G(u)|=\max_{0\leq u\leq1}|\mathbb G(u)|.
$$
We obtain the <Kolmogorov-Smirnov theorem> in the required form:
$$
\boxed{\sqrt n\sup_{t\in\mathbb R}|F_n(t)-F(t)|\xrightarrow d Z,\qquad Z=\max_{0\leq u\leq1}|\mathbb G(u)|.}
$$
Continuity of $F$ is what makes the limit independent of the unknown <cumulative distribution function>.
Choose $c_\alpha$ with $\mathbb P(Z\leq c_\alpha)=1-\alpha$. The law of $Z$ has no atoms. For example, the <Brownian reflection principle> and conditioning a <Brownian motion> on its endpoint give $\mathbb P(\max\mathbb G>a)=e^{-2a^2}$ for $a>0$: the reflected endpoint density at zero is the normal density at $2a$, divided by its value at zero. Hence both one-sided maxima have continuous laws, and an atom of $Z$ would require an atom of one of them. Also $\mathbb P(Z=0)=0$, since $\mathbb G(1/2)$ is nondegenerate. Thus such quantiles give the desired limiting coverage.
An asymptotic simultaneous <confidence band> is
$$
\boxed{L_n(t)=\max\{0,F_n(t)-c_\alpha/\sqrt n\},\qquad U_n(t)=\min\{1,F_n(t)+c_\alpha/\sqrt n\}.}
$$
Indeed, because $0\leq F\leq1$, the event that $L_n(t)\leq F(t)\leq U_n(t)$ for every $t$ is precisely $\sqrt n\|F_n-F\|_\infty\leq c_\alpha$. Its <probability> tends to $1-\alpha$. This is simultaneous coverage over all real $t$, rather than separate pointwise <confidence intervals>.
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