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# difference in sup and inf of $f(x)$ is less than $L_1$ norm

2017-12-03 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 $\sup_K -\inf _K = 1-0=1$.

• 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 $\sup_K -\inf _K = 1-0=1$.

2017-12-03 03:26:12