Qu1. SupposeA is a function that assigns to each subset of the plane a number in such a way that the following rules are satisfied:

1:A(X)0 for eachXR2

2:IfX andY are disjoint thenA(XY)=A(X)+A(Y)

Prove that ifXY thenA(X)A(Y)

Qu2.(i) Suppose thatF is an increasing function andF(0)>0. Define G(x)=0xF.Prove that G is one-to-one.

(ii)LetSbe the sawtooth function defined by

S(u)={uu1if0u1if1<u2

Prove thatS is Riemann integrable.(Hint:S is almost a piecewise monotone function, but not quite!)

(iii)LetT(x)=0xS whereS is the sawtooth function from the previous exercise. Prove thatTis not differentiable at 1. Why does this not contradict the Fundamental Theorem of Calculus?

3(i).Find a pair of functionsf andg and an interval[a,b]such that

(abf)(abg)=abfg .

(note the sign in this integral equation was suppose to be not equal to, dont know why its not working in latex.)

(ii)Prove that for any Riemann integrable functionf on the interval[a,b] there isc such thatacf=cbf .