TMA4225: Exercises for week 45

131X(a), 132X(a), 225X(b) (recall that Fremlin's “\(gf\)” is the composition, more commonly written “\(g\circ f\)”).

If \(f\colon[a,b]\to\mathbb{R}\) is absolutely continuous, let \(V(x)\) be the variation of \(f\) on \([a,x]\). Show that \(V\) is absolutely continuous.

Show that \(V(y)-V(x)\ge|f(y)-f(x)|\) when \(x<y\), and conclude that \(V'\ge |f'|\) a.e.

Show that \(V(y)-V(x)\le\int_x^y|f'|\) when \(x<y\), and conclude that \(V'=|f'|\) a.e. Finally, show that \[\mathop{\rm TV}f=\int_a^b |f'|.\]