If I have a function $g:[0,1]\rightarrow [0,1]$ which is continuous and $f:\Omega \rightarrow [0,1]$ which is right continuous. Is $g(f)$ continuous or just right continuous? Here $\Omega$ is the event space and $f$ is a random variable.
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In general it will only be right-continuous as we can take $g(x)=x$ and then $g\circ f=f$.
Severin Schraven
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any formal way to prove this property? – mijsi kio Oct 30 '20 at 17:01
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That is a formal proof. Just pick any function $f$ that is right-continuous and not continuous, then $g \circ f = f$ is right-continuous and not continuous. – Severin Schraven Oct 30 '20 at 17:02
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Or which property do you mean? – Severin Schraven Oct 30 '20 at 17:05
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yeah got it thanks – mijsi kio Oct 30 '20 at 17:11
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You're welcome. – Severin Schraven Oct 30 '20 at 17:15