# $f:[0,\infty)\to \mathbb{R}$ right continuous with left limits. Then is $\{y:\lambda(t:f(t)=y)=0\}$ dense in $\mathbb{R}$?

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.

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.