Suppose f:[0,1]->R is absolutely continuous and |f'(x)|<M almost everywhere in [0,1]. Prove that f is Lipschitz.
I wrote up the following proof and got significant deductions:
By the Mean Value Theorem, for all x,y in [0,1], there exists c between x and y such that...
Hi everyone,
I have a general question on choosing a subfield within pure mathematics.
I personally find Analysis (more specifically, functional analysis) interesting, engrossing, and fun. I also find that I am better at it than, say, algebra.
On the other hand, I can't help but feel like...
What do people think about Lang's textbooks?
I am learning Complex Analysis for the first time from Lang. (literally the 1st time I'm seeing ANY sort of complex analysis/calculus, although I have extensive experience with real analysis).
I have gone through quite a love-hate relationship with...
In R, every nonempty open set is the disjoint union of a countable collection of open intervals. (Royden/Fitzpatrick, 4th edition)
What is the most general setting in which every open set is a disjoint union of countable collection of open balls (or bases)? In R^n? In metric spaces? In second...
What is the best way to intuitively and visually distinguish between immersion and submersions? For example, I understand that the standard picture of the Klein Bottle in R^3 is an immersion. How do I see this? (Obviously, it's not an embedding because the Klein Bottle self-intersects in R^3...
Let S be an uncountable subset of the reals.
Then does S always contain at least one interval (whether it be open/closed/half-open/rays/etc..)?
Maybe the Cantor set is an example of an uncountable set that contains no intervals? How does one show this if it is true?
My intuition is...
Nice! It didn't occur to me to think in terms of functions. But how does "multiplying both sides by x" work? Because doing that sometimes introduces extraneous roots. (For example, multiply x=1 with x on both sides). But f(A)=xA is normally considered an injective function.
And using the term...
Hello,
I am a first-year math grad student, and I just started teaching two sections of precalc! yikes!
So I had a major embarrasing moment during office hours when I told 2 students the wrong answer because I completely forgot about the phenomenon known as "Extraneous roots"! Does anyone...
Is there a proof that shows if R^n can be turned into a field, for specific n?
Obviously, n=1 is a field, and n=2 can be made into a field (which is just the complex plane.)
So what about n>2?
Hi everyone,
Does anyone have any recommendations on good textbooks (or websites) that help translate the mathematicians' language of tensors (strictly as multilinear maps to the underlying field) and forms (as alternating tensor fields) to the language used by physicists?
For example, I want...