Suppose f: [a,b] $\to$ $\mathbb{R}$ and B satisfy |f(x)| $\le$ B for every x $\epsilon$ [a,b].

Show that if P = {x$_{0}$,...,x$_{n}$} is a partition of [a,b], then

M(f$^{2}$,[x$_{i-1}$,x$_{i}$]) - m(f$^{2}$,[x$_{i-1}$,x$_{i}$]) $\le$ 2B(M(f,[x$_{i-1}$,x$_{i}$]) - m(f,[x$_{i-1}$,x$_{i}$]))

for every 1 $\le$ i $\le$ n.

We are given a hint, namely that

|f(x)$^{2}$ - f(y)$^{2}$| = |f(x) - f(y)||f(x) + f(y)|.

And this has something to do with Riemann integrals, or perhaps Darboux sums, as that is the section this homework was assigned in.

  • $\begingroup$ What are the functions $M$ and $m$? $\endgroup$ – Sean Roberson Nov 22 '18 at 2:51
  • $\begingroup$ Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets. $\endgroup$ – user610107 Nov 22 '18 at 2:55
  • 4
    $\begingroup$ The result is obvious from the hint and triangle inequality $|f(x) +f(y) |\leq 2B$ and $|f(x) - f(y) |\leq M_f-m_f$ $\endgroup$ – Paramanand Singh Nov 22 '18 at 4:18

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