Prove the following inequality (Oscilations)

Question

For $f,g: \mathbb{R}^n\to [-M,M]$,

Prove: $Osc_{fg}\leq M(Osc_f+Osc_g)$

Where $Osc_f(U)= supf(x)-inf f(x)$ (for $x\in U\subset\mathbb{R}^n)$

I tried using the identity $4fg=(f+g)^2-(f-g)^2$ but I seem to be stuck.. Can anyone help?

Answer 1

The following definition of Osc is often useful (because it avoids problems with signs later on):

$${\rm Osc}_f = \sup \{|f(x)-f(y)|: x,y \in U \}.$$

And then,

$$ |f(x)g(x)-f(y)g(y)| \leq |f(x)(g(x)-g(y))| + |(f(x)-f(y))g(y)| \leq M {\ \rm Osc}_g + M {\ \rm Osc}_f $$