Recent content by blinktx411

the key to this problem is in the first line: "for all x > 0". Log(x) is not defined for any x ≤ 0

Try drawing the region and switching the order of integration, and this might allow you to use substitution rule hopefully, as I believe you will get something of the form \int y*\sin(\pi y^2) dy

Thanks for the reply! This is what I've been trying, but I cannot see a reason why |f_n-f|^t should be bounded on this set.

I'd like to note how close f=|x| comes to breaking this! So, I can get that the function isn't continuous, and perhaps this will help and perhaps it won't. Suppose it were continuous, and let y be in the range of f. Then consider I=[x_1,x_2] where f(x_1)=f(x_2)=y. Now, on I f attains a max, say...

\rho (x,y) = |x|

i did this because in the problem statement you said rho = |x|. But yes, this looks to be correct.

well your bounds should be pretty easy, theta goes from 0 to pi and r goes from 0 to 1. Your question of rho = |x| comes down to knowing x = r cos(theta) and |r cos(theta)| = r |cos(theta)|, which is just a question of knowing where cos(theta) is positive and where it is negative.

It looks like you're integrating over a rectangular region in your solution, but the region given is the upper half of a disk. Also, polar coordinates seem like they would be more friendly here.

Do you know the fact that for every real number y there is a rational r such that |r-y| < epsilon for any epsilon > 0? Or, in a similar fashion, for any real number y there is a sequence of rationals converging to it? Perhaps these facts will help.

Homework Statement I have a sequence of functions converging pointwise a.e. on a finite measure space, \int_X |f_n|^p \leq M (1 < p \leq \infty for all n. I need to conclude that f \in L^p and f_n \rightarrow f in L^t for all 1 \leq t < p. Homework Equations The Attempt at a Solution...

Here's a hint: k^(1/2) > ln(k) for all k.

Why do you think the original series is divergent? Your analysis looks correct.

Hello PF! Was wondering if anyone knew a good reference on the topological characterization of the cantor set, proving that if a metric space is perfect, compact, totally disconnected it is homeomorphic to the cantor set. Thanks!

Currently in my course in topology we have covered the point-set portion of the Munkres text, and the professor has moved into some additional material in which munkres has no resources, mainly the classification of surfaces. The professor let me borrow his resource for a while, but I was...

Thank you! The problem is from bartle, 27s if you're interested. The context suggests I should be able to prove this from MVT directly, but so far no luck. I'm going to take a closer look at your suggestion, and thanks!