spetharrific askedWhoops……maybe. I’ll check my notes on that one.
Edit: No, the function is defined as a over n, where n is the number of rationals less than x on the interval [0,1].
Since the rationals are countable, let them be listed as such:
Now, for x in [0,1], let f(x) be:
Is this monotone? Is this continuous? If so, where? Is this Riemann integrable on alpha = x? If so, prove it. Bonus points for a 2 line dis/proof of the integrability.
Ps — You can cite both Krantz and (Baby) Rudin. I have copies of both one hand.
I had to finish up my thesis and defend it (succeed, too!)
And then take finals, the last of which was today.
So, as of Saturday (assuming I pass my classes), I will have my Master’s.
I also have 2 job interviews for Friday and a third for Monday.
Is this what heaven feels like?
Some of you may remember awhile back, I posted about how Conservapedia thought that Jesus used Banach-Tarski to feed the hungry.
Well, now, they’re claiming Relativity is not only false, but a liberal conspiracy by Obama to help abort babies……go read for yourself.
Some highlights:
It is unlikely tenure or a Ph.D would be awarded to any critic of the theory.
Prove it wrong, numbnuts.
Despite censorship of dissent about relativity, evidence contrary to the theory is discussed outside of liberal universities.
Yup, relativity is alllllllllll about liberalness.
I think the biggest part of hilarity is that the have relativity and quantum mechanics mixed.
Creation scientists such as physicists Dr. Russell Humphreys and Dr. John Hartnett have used relativistic time dilation to explain how the earth can be only 6,000 years old even though cosmological data (background radiation, supernovae, etc.) set a much older age for the universe.
First you say that Relativity isn’t right, but it’s right when you use it for that? Pick a side, buddy.
The data selection could be considered “manipulation” or “fudging”, by a person (Arthur Eddington) who had a personal stake in the outcome. His analysis techniques would not pass muster today.
Yup, because things are exactly how they were 100 years ago, including even in our calculations. Oh, wait, these people think hate crimes are a ‘liberal invention’.
Bonus: It includes links to the page for ‘Action at a distance’, which includes “Action at a distance performed by Jesus, as described in John 4:46-54 and Matthew 27:51.”
Consider this cringeworthy guy:
Tell me the limit:
Hint:
The top one is bounded, the bottom one unbounded and monotonic increasing.
Good luck!
Zero. sqrt(n) -> ∞ as n -> ∞. sin(nx) -> +/- 1 as n -> ∞.
Bingo!
Consider this cringeworthy guy:
Tell me the limit:
Hint:
The top one is bounded, the bottom one unbounded and monotonic increasing.
Good luck!
Wow. Looks daunting enough, don’t it? Holy hell, that’s evil. What could be worse?
Eep.
So, how we gonna do this? Well, we know that:
because
This holds for x=1, as well, since then it’s just values of multiples of pi in the argument afterwards.
In fact, if you play around with it, noting that m! has the product of every value of m up to it in it, then any x which has some integer <= m as the denominator would make the cosine argument 0. So, we can say that as m goes to infinity, any rational number would make it 1. So the limit is 1 over the rationals.
So, what if x is irrational? Then we see that the limit will be less than 1; then, we get a decreasing monotone function which goes to 0.
Hence, the limit function is identically 0 over the irrationals and identically 1 on the rationals.
It’s also not continuous, not differentiable (well, duh, it’s not continuous), and it’s not integrable at all.
Science-ify your breakfast with a Möbius bagel
You’ve probably heard of a Möbius strip before - it’s a continuous shape that only has one side and one edge. You can make one pretty easily by cutting a strip of paper, giving it a half twist, and taping the ends together to form a loop. However, if you want to really impress, make a Möbius bagel. By following the instructions, you can cut your bagel (or your donut!) into two interlocking bagel halves. From now on, eat your breakfast like a Scientist!
