$\phi: K \to R$
K compact, $\phi$ is $C^0$
$f: [a,b] \to K$
Then $\phi \ o \ f$ is RI.
I do know that the composition of two integrable and continuous functions is integrable. In this problem, I know $\phi$ is RI because it is continuous. However, all I know about $f$ is that it maps a compact interval $[a,b]$ to a compact space $K$. I am not told it is continuous nor that it is RI. So I believe my question is:
Is a continuous function (therefore RI function) composed with a not necessarily continuous nor RI function still RI? And how would I prove that? Any help is appreciated! Thanks!