EDB — 1BT

Exercises

1. [1BT] Prerequisites:[18F]↺↻, [1BF]↺↻.Let \(I⊂ ℝ\) be an open interval. Suppose that \(g:I→ℝ\) in Riemann integrable on any bounded closed interval contained in \(I\). Fixed \(x,y\in {\mathbb {R}}\) with \(x≠ y\), let

   \[ R(x,y )=\frac 1{y-x}∫_ x^ y g(s)\, {\mathbb {d}}s \]

   (with the usual convention that \(∫_ x^ y g(s)\, {\mathbb {d}}s=-∫_ y^ x g(s)\, {\mathbb {d}}s\), so that \(R(x,y)=R(y,x)\)). If \(g\) is monotonic, show that \(R(x,y)\) is monotonic in each variable. If \(g\) is continuous and \(R(x,y)\) is monotonic in each variable, show that \(g\) is monotonic.

   Solution 1

   [1BV]↺↻