I would like to understand better how Skorokhod's $J_1 $ topology works in practical matters and I thought that proving continuity of some functionals would be a good place to start. Just to make sure that I am clear, I am considering the space $D[0, 1] $ of real-valued cadlag functions defined on $[0, 1]. $ The definition of Skorokhod's $J_1 $metric is the following: $$ d_S(f, g) = \inf_{\lambda \in \Lambda}\bigl\{\epsilon > 0: |\lambda(t) - t|, \le \epsilon; |f(t) - g(\lambda(t))| \le \epsilon\bigl\} $$ and $\Lambda $ is the class of all homeomorphisms $\lambda: [0, 1] \mapsto [0, 1] $ such that $\lambda(0) = 0 $ and $\lambda(1) = 1. $
There are several properties for this metric (i.e, separable, but not complete). However, at this stage all I would like to do is to show myself how the metric works. Continuity is usually a good point to start. So I thought that
proving that the functional
$$F(f) = \sup_{t\in [0, 1]} f(t), \mskip 4pt f \in D[0, 1] $$
is continuous would be a reasonable place to start my investigation. For that I need to prove that $F $ is continuous at $f_0
\in (D[0, 1], d_S) $ for each $f $ in a certain neighborhood of $f_0. $
So I started looking at
$$ |F(t) - F(t_0)| < \epsilon $$
Now,
$$|F(t)-F(t_0)| = |\sup_t f(t) - \sup_t f_0(t)| \le \sup_t |f(t) -f_0(t)| $$
which I could further write:
$$\le \sup_t |f(t) - f_0(\lambda(t))| + \sup_t|f_0(\lambda(t)) - f_0(t)|. $$
The first term in the last formula can be made less than $\epsilon $ by choosing $f \in B_{d_S}(f_0, \epsilon). $ However, I do not
see how to show that the term $\sup_t|f_0(\lambda(t)) - f_0(t))| $ is also less than $\epsilon. $ It is true that $d_S(f_0, f_0) = 0, $ but the homeomorphism $\lambda $ to use for computing the distance could be different (in fact in this case we need to choose $\lambda(t) = t $).
Am I missing something that or, perhaps, am I wrong to think that $F(f) = \sup_{t\in [0, 1]} f(t) $ is a continuous functional in $D[0, 1] $ equipped with Skorokhod's $J_1 $ topology ?
Thank you in advance to anyone who can help me see what I am missing.
Maurice