11
votes
3answers
523 views

Why study metric spaces?

Most universities have a 3rd year undergraduate analysis course in which metric spaces are studied in depth (compactness, completeness, connectedness, etc...). However, in practice it seems that most ...
3
votes
1answer
126 views

What are the benefits or losses of learning real analysis through a constructivist approach instead of a standard apporach?

Recently I've found some courses on real analysis that use the constructivist approach and I got curious on some aspects: What are the benefits of learning through this approach? Is it ok to learn ...
3
votes
0answers
74 views

A question on mathematical writing.

One of the problems I am grading this week is as follows: Given a simply connected bounded domain $\Omega$ on $\mathbb{R}^{2}$, prove that there exist a line that separates it into two parts of equal ...
6
votes
4answers
566 views

How to convince a high school student that differentials don't work like fractions in general?

It all started when I tried to convince a 10th grader that if $f$ is a function defined on $\mathbb{R}^n$ the differential is defined by: $\large \displaystyle df = ...
6
votes
1answer
126 views

The Value of a series

What is the value of the following series $\sum_{n=1}^\infty\sum_{m=1}^\infty\sum_{k=1}^\infty \frac{1}{mnk(m+n+k+1)}$
5
votes
1answer
380 views

The Constant Function Theorem first of all $\,$?

I quote Thomas W.Tucker $\,$ "... By the way, I view the Constant Function Theorem as even more basic than the IFT. It would be nice to use it as our theoretical cornerstone, but I know of no way to ...
2
votes
0answers
278 views

These unknown uniformly differentiable functions

Let $f$ be defined on $[a,b]$ and there uniformly differentiable ($\,$the $\delta$ in the definition of derivative is independent of the point). Given $\epsilon>0$, choose a partition $P \, : \, ...
5
votes
1answer
307 views

How does one visualize a function with a discontinuous second derivative?

Let us assume that all functions are continuous. I was teaching my calculus students the other day. We were talking about what points of non-differentiability look like. Two ways a function can fail ...
47
votes
3answers
3k views

Getting Students to Not Fear Confusion

I'm a fifth year grad student, and I've taught several classes for freshmen and sophomores. This summer, as an "advanced" (whatever that means) grad student I got to teach an upper level class: Intro ...
19
votes
8answers
3k views

Why do introductory real analysis courses teach bottom up?

A big part of introductory real analysis courses is getting intuition for the $\epsilon-\delta$ proofs. For example, these types of proofs come up a lot when studying differentiation, continuity, and ...
2
votes
1answer
105 views

weakly locally one-to-one?

Is there any standard name for this concept that is weaker than local one-to-one-ness? In some open neighborhood of $x_0$ there is no point $x\ne x_0$ such that $f(x)=f(x_0)$. Or, if you like: In ...