Suppose we have a function $$ f(x) = 1-\alpha F^-(x) $$ Where $F^-$ is the left limit of a CDF.
In general is $f(x)$ not continuous? Is it possible for $f(x)$ to be continuous if $F$ is not?
What if $F$ is continuous (so $F(x) =F^-(x))$?. Then $f(x)$ should be continuous as the composition of continuous functions correct?
I guess my confusion is that $F^-$ is not $F$. hence I don't know how I should "think about" this function.
That is, if $F$ is discontinuous it has jumps (or maybe some other weirdness, but maybe not because CADLAG, I'm not sure) so in general $g(x) = 1-\alpha F(X)$ is not continuous. But I don't know if $F^-$ has jumps (or some other weirdness), or if I am overlooking something