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Results tagged with mathematical-analysis
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user 5097
For questions applying to analysis courses: Real and complex analysis. Typically a higher and more proof-based level than calculus.
13
votes
6
answers
5k
views
Why should we study continuity?
This question is related to How can I motivate the formal definition of continuity? Imagine a student asks the question why it is worth it to study continuity. What is a good response to this question …
- 2,031
2
votes
1
answer
142
views
Proving convergence or divergence of series: tips and recommendations
This is a follow up of my question on MSE. Which tips and recommendations would you give students who want to investigate series about convergence or divergence?
So far we have collected:
It is go …
- 2,031
4
votes
1
answer
1k
views
Why is continuity defined as a local property?
The formal definition of continuity is a local property (the definition of continuity at a point is a property of the germ of the function at this point). Why is it a good decision to make the definit …
- 2,031
4
votes
2
answers
1k
views
Why is continuity only defined on its domain?
As mentioned in this question students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \to \mat …
- 2,031
12
votes
9
answers
12k
views
Why do we study ordinary differential equations?
What is a good answer to the question: Why should one study ordinary differential equations?
I would give the answer: ODEs are used in many models to determine how the state of this model is changi …
- 2,031
20
votes
7
answers
2k
views
How can I motivate the formal definition of continuity?
In order to teach continuity of real valued functions $f:D\to\mathbb R$ one may start with the (in some sense wrong) intuition
$f$ is continuous when its graph can be drawn without lifting the pen …
- 2,031
8
votes
6
answers
2k
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What is the intuition behind the limit superior?
I want to write an article which explains the limit superior. I also want to present the intuition behind this concept. Currently I would describe the limit superior as the "least upper bound of a seq …
- 2,031