1. ## Integration

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

|integral(f*gdx)|^2<=integralf^2dx*integra1g^2dx

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.

2. Originally Posted by kathrynmath
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

|integral(f*gdx)|^2<=integralf^2dx*integra1g^2dx

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.
I'm sorry, but this is near unreadable. You need to latex this.

3. 1.Let f be integrable on [a; b] for every b > a; where a is fixed. Define

provided the limit exists.

Prove the so called integral test: if f(x) >= 0 and if f decreases monotonically for x >=1, then converges (to a finite number) if and only if 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:

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.

4. for 2, I have integral(f+tg)^2
integral(f^2+2ftg+t^2g^2)

5. Ok I got 1 and 2 fairly well figured out. Any hints on starting 3 and 4?