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20171203 03:06:49
Is the following true for a function $f(x)$ defined over a compact set $X$?
$\sup f(x)  \inf f(x) \leq \int f(x) dx$
Do we need the function to be continuous?
No, it's not true, not even for continuous $f$. Consider $f(x)=x^n$ for $x\in K=[0,1]$. The integral is $1/(n+1)$ but $\sup_K \inf _K = 10=1$.

No, it's not true, not even for continuous $f$. Consider $f(x)=x^n$ for $x\in K=[0,1]$. The integral is $1/(n+1)$ but $\sup_K \inf _K = 10=1$.
20171203 03:26:12