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Results tagged with real-analysis
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user 81488
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
5
votes
2
answers
1k
views
An example of an open discontinuous function
Consider the following simple example of a function $f: \mathbb{R}\to\mathbb{R}$ which is open and discontinuous at all points. If $x\in\mathbb{R}$ is represented as something.$x_1x_2x_3\dots$ in the …
- 1,897
4
votes
In the plane, does complement of Brownian path have infinitely many connected components?
Yes. Just observe that
(1) on any fixed time interval the Brownian path intersects itself with positive probability (easy to see);
(2) but the above implies that on any time interval the Brownian pa …
- 1,897
4
votes
Slight variation on law of the iterated logarithm
You can use the fact that $B_t-m_t$ is a reflected Brownian motion (see e.g. Revuz-Yor, Chapter VI, Theorem 2.3). I think it shouldn't be difficult to show that
$$
\limsup_{t\to\infty} \frac{M_t-m_t}{ …
- 1,897
2
votes
How many steps do I have tto complete? Recursive sequence
You may find it useful to look here: http://www.artofproblemsolving.com/community/c7h398470 (in particular, the link in the 1st post, and also fedja's answer).
- 1,897