1.Let f be integrable on [a; b] for every b > a; where a is fixed. Define
integralf(x)dx(bonds on integral(a,infinity) = limb-->infinity(integralf(x)dx)(bounds a,b);
provided the limit exists.
Prove the so called integral test: if f(x) >= 0 and if f decreases monotonically for x >=1, then integral f(x)dx(bounds 1,infinity) converges (to a finite number) if and only if sum(f(n)converges.
2. Let f and g be integrable functions on [a; b]: Prove
3.Assume f is integrable on [a,b] and has a jump discontinuity at c in (a,b). This means that both one sided limits exists as x approaches c from the left and right, but that they are not equal.Show that F(x)=integral(f)(bounds a,x) is not differentiable at x=c.
4. The existence of a continuous monotone function that fails to be differentiable on a dense subset of R is what this problem concerns. Show how to construct such a function using the result of the previous problem and the fact that h(x) is a monotone function defined on all of R that is continous at every irrational point where:
h(x)=sum(u_n(x)) u_n(x)=1/2^n for x>r_n and 0 for x<=r_n
where r_n are the rational numbers
Here are my ideas so far:
1. I feel like the monotonically decreasing part will help.
2. For 2, I considered the quadratic polynomial in the variable t defined by P(t) =integral(f + tg)^2. I squared this all out which worked out fine and I feel like this will help me, but I don't know what to with the t's.
3. I feel like the answer comes from the fact that the limits are not equal and I use that somehow.
4. Since I haven't figured out 3, have no clue.