Let $f:[0,\infty)\to \mathbb{R}$ right continuous with left limits. I believe it is true that $\{y:\lambda(\{t:f(t)=y\})=0\}$ is dense in $\mathbb{R}$, where $\lambda$ is the Lebesgue measure. This fact reminds me of another property of rcll-càdlàg functions, that is that the set of continuity points is dense in $[0,\infty)$; however, I can't find a proof of this either. I would really appreciate a sketch or hint to prove/disprove these (kinds of) facts because I wouldn't know how to construct an argument for these.
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I think your first statement doesn't rely on the cadlag property.
Set $A_n=\{y:\lambda(\{t:f(t)=y\}\cap [0,n])>0\}$. Note that $A_n$ can be at most countable. Indeed, suppose $A_n$ was uncountable. Then, there exists $m>0$ so that the set $A=\{y:\lambda(\{t:f(t)=y\}\cap [0,n])>\frac{1}{m}\}$ is infinite. But then $ \cup_{y\in A}\{t:f(t)=y\}\cap[0,n] \subset [0,n] $ and the left set has infinite measure by the argument above. Hence, the set $\cup_{n \in \mathbb{N}}A_n$, it is at most countable. This is just the complement of your set. But any countable set is meagre, and hence has dense complement by the Baire Category Theorem.
Regarding the density of continuity points, this can once again be shown by a countability argument, see for instance this post.
