Math 325 Elementary Analysis Fall 2016 University of Nebraska-Lincoln Assignment #8 Hard deadline: Tuesday, December 6 at 10:30am Will be graded out of 60 points (20 points for each problem) 1 Definitions Given a bounded function f : [a, b] ! R, the total variation of f over [a, b] is defined by ( n )! X V[a,b] (f ) = lub |f (xk ) f (xk 1 )| : a = x0 < x1 < x2 < < xn = b . k=1 A function f : [a, b] ! R has bounded variation over [a, b], if V[a,b] (f ) < 1, and the space of functions of bounded variation (BV functions) is BV([a, b]) = f : [a, b] ! R : V[a,b] (f ) < 1 . 2 Problem Set 1. For this problem, define f, F : R ! R by f (x) = ( x, 1 x<0 2