The set $C[0,1]$ is nowhere dense in $D[0,1]$.

Question

$D[0,1]$ be the space of real functions $x$ on $[0,1]$ that are right continuous and have left hand limit.

On the other hand $C[0,1]$ is set of all continuous real functions on that interval.

Considering the underlying topology is Skorohod topology. I try to show that the set $C[0,1]$ is nowhere dense in $D[0,1]$. How to do that?

One idea is,

Suppose consider the ball in $D[0,1]$ i.e.
$B(f,\epsilon) = \{g\in D[0,1] | d(f,g) < \epsilon\}$, where $d$ is Skorohod metric. Now it is enough to show that
$C[0,1] \cap B(f,\epsilon) = \emptyset$. Then we can conclude that $C$ is nowhere dense.

But how to show this claim?

Answer 1

An explicit development.

Claim 1. Let $g\in D\left[0,1\right]$ be a càdlàg function with a jump of size $J>0$ at $t\in \left(0,1\right)$, i.e., $J=\left|g(t)-\lim\limits_{x\rightarrow t-} g(x)\right|$. Then, $$\|h-g\|\geq \frac{J}{2}\tag{$\star$}\label{star},$$ for any $h\in\mathcal{C}\left[0,1\right]$.

Proof. Remark that $|h(t)-g(t)|+|h(t)-\lim\limits_{x\rightarrow t-} g(x)|\geq J/2$. Consider $\delta>0$ small enough so that i) [h is continuous] $|h(x)-h(t)|<\varepsilon/3$ for all $x\in\left(-\delta,\delta\right)$; ii) [$g$ is right-continuous] $|g(x)-g(t)|<\varepsilon/3$ for all $x\in\left(0,\delta\right)$; and iii) [$g$ has left-limit] $|g(x)-\lim\limits_{x\rightarrow t-} g(x)|<\varepsilon/3$ for all $x\in\left(-\delta,0\right)$. Then, $|g(x)-h(x)|\geq J/2-\varepsilon$ for all $x\in \left(-\delta,\delta\right)$. In other words, $\|h-g\|\geq J/2-\varepsilon$ for any $\varepsilon>0$ and thus, $\|h-g\|\geq \frac{J}{2}$.

Claim 2. Let $g\in \mathcal{D}\left[0,1\right]\setminus \mathcal{C}\left[0,1\right]$ be a discontinuous càdlàg function -- i.e., it has a positive jump $J=\left|g(t)-\lim\limits_{x\rightarrow t-} g(x)\right|>0$ for some $t$. Then, $\mathcal{C}\left[0,1\right]\cap B\left(g,r\right)=\emptyset$ for any $r<J/2$, where $J$ is the size of one of the jumps of $g$.

Proof. Let $f\in \mathcal{C}\left[0,1\right]$. Observe that: $$d(f,g)=\inf_{\lambda\in \Lambda} \left\{ \|\lambda-I\|\vee \|f\circ\lambda - g\| \right\}\geq \inf_{\lambda\in\Lambda}\|f\circ\lambda - g\|.$$

Since $f\circ \lambda$ is continuous for any $\lambda\in\Lambda$ then, from Claim 1., we have that
$$\|f\circ\lambda - g\|\geq \frac{J}{2}\,\,\,\forall{\lambda\in\Lambda}.$$ Therefore, $d(f,g)\geq \frac{J}{2}$, for any $f\in \mathcal{C}\left[0,1\right]$. In other words, $\mathcal{C}\left[0,1\right]\cap B(g,r)=\emptyset$ for any $r<J/2$.

Claim 3. $\mathcal{D}\left[0,1\right]\setminus\mathcal{C}\left[0,1\right]$ is dense in $\mathcal{D}\left[0,1\right]$.

Proof. Given any $f\in\mathcal{C}\left[0,1\right]$, just consider the sequence $g_n(x):= f(x)+(\frac{1}{n})1_{\left\{x\geq t\right\}}$ and clearly $g_n\in\mathcal{D}\left[0,1\right]\setminus \mathcal{C}\left[0,1\right]$ for all $n$. We have $\|g_n-f\|\overset{n\rightarrow \infty}\longrightarrow 0$ (which also implies convergence w.r.t. the Skorohod metric).

Answer to the question. Since $\mathcal{D}\left[0,1\right]\setminus\mathcal{C}\left[0,1\right]$ is dense in $\mathcal{D}\left[0,1\right]$ (Claim 3.), the result follows: i) any (nonempty) open set $\mathcal{O}$ in $\mathcal{D}\left[0,1\right]$ contains a discontinuous $g$; ii) $B(g,J/3)$ does not contain any continuous function (Claim 2.), where $J>0$ is a jump of $g$. Therefore, $\mathcal{C}\left[0,1\right]$ is not dense in $\mathcal{O}$. This concludes the proof.