The oscillation of $\omega_f(A)$ of $f$ on a set $A$ to be the number $\omega_f(A)=\sup\limits_{x,y\in A}|f(x)-f(y)|=M_A(f)-m_A(f).$
The following equality is where I'm scratching my head a bit: $\sup\limits_{x,y\in A}|f(x)-f(y)|=M_A(f)-m_A(f).$
Where $M_A(f)=\sup\limits_{x\in A} f(x)$, $m_Af(x)=\inf\limits_{x\in A} f(x)$, and $f$ is a bounded function on $A$.
So here is my attempt to prove the equality. Putting how we defined $M_A(f)$ and $m_A(f)$ together: $M_A(f)-m_A(f)=\sup\limits_{x\in A} f(x)-\inf\limits_{x\in A} f(x).$
Since $f$ is bounded: $\inf\limits_{x \in A} f(x)=-\sup\limits_{x \in A} -f(x)$.
So, $M_A(f)-m_A(f)=\sup\limits_{x\in A} f(x)+\sup\limits_{x \in A} -f(x).$ $M_A(f)-m_A(f)=\sup\limits_{x\in A} f(x)-\sup\limits_{x \in A} f(x).$
I'm wondering if this is right so far or if I've drifted off to far left field.