
 $
 \newcommand{\essSup}{\mathop{\rm sup\_{ess}}\nolimits}
 $


What can be concluded from the fact, that $f: X\ni x\mapsto f(x)\in [a,b]\subset\mathbb{R}\setminus\lbrace{-\infty,+\infty\rbrace} $ has the following property: 


$$ f(x\_0) = \lim\_{\epsilon\to 0} \essSup\left(f\left(x\in X\cap U\_\epsilon\left(x\_0\right)\right)\right)$$ 



> 
> **Questions:**
> 
> 
> * are such functions Riemann integrable, and if not,
> * what are examples of such functions, that are not Riemann integrable?
> 
> 
> 



Implications of a Regularity Condition for Functions
Example: the characteristic function of a fat Cantor set (a compact nowhere dense set of positive Lebesgue measure): it is upper semicontinuous, so the equality holds; it has a positive set of discontinuities, thus it is not Riemann integrable.


