The Median of a Continuous Function
Sunny Garlang Noah · Real Analysis Exchange · 2008
Let $\Omega $ be a domain in $\mathbb{R}^{n}$ with finite Lebesgue measure and $f\in C\left( \Omega \right) \cap L^{1}\left( \Omega \right) $ a real-valued function on $\Omega $. It is shown that there exists a unique number $M\in \mathbb{R}$ at which the function $I\left( y\right) =\int_{\Omega }\left| f\left( x\right) -y\right| d\lambda ^{n}\left( x\right) $ is minimized, where $\lambda ^{n}$ is the Lebesgue measure on $% \mathbb{R}^{n}$. We can define this number as \textit{the }median of $f$ over $\Omega $ with respect to $\lambda ^{n}$.