Question

The function,f: [-1,1]×[2,1] → R, defined by

f(x,y)= { x ; y is rational

{ 0 ; y is irrational

is integrable or not.

True or false with full explanation

Answer 1

$Inf U (p, f) =sup L(p, f)$

$L(p, f) =\sum_(i=1) mi\Delta xi$

$L(p, f) =m1\Delta x1+m2\Delta x2 +... +mr\Delta xr +... +mn\Delta xn$

$mi=0 \therefore L(p, f) =0$

$Sup L (p, f) =0$

$U(p, f) = \sum _(i=1) Mi\Delta xi$

$U(p, f) =M1\Delta x1+M2\Delta x2+... +Mr\Delta xr+... +Mn\Delta xn$

$M1=0, Mr =1$

$U(p, f) = 1\Delta xr$

$\Delta xr\to 0 \therefore inf U(p, f) =0$

Hence integrable