Thanks for the reply!
But I am afraid I still don't see how this can be used to show that f is constant.
I think you are going for the fact that f must be constant since it is arbitrarily close to a constant function along every sequence?
There must be something I'm missing. It's the...
Hello,
I have a problem I cannot solve. I have been working with problems with convergence of sequences of functions for some time now. But I can't seem to solve most of the problems. Anyway here is my problem:
Consider a continuous function f: [0, \infty) \rightarrow \mathbb{R} . For each...
Hi there!
Is the following true?
Suppose A is an open set and not closed. Cl(A) is closed and contains A, hence it contains at least one point not in A.
If A is both open and closed it obviously does not hold.
Thank you again.
I believe I have got it right now.
Let S_0 be the set of points which can be joined to x_0 \in U by a path in U . Now for any y \in S_0 there is some open ball B_{\epsilon}(y) fully contained in S_0 . This is true because by our definition of S_0 , there is...
Thank you micromass.
WannabeNewton: I found my error and changed it from "And U is connected, which implies that the intersection of any two balls, can not be empty" to "And U is connected, which implies that given any ball, there is another ball with nonempty intersection"
But I just...
Thanks for the replies!
Before I post my reply, how do I write math? To the right it says "Click the sigma symbol in the toolbar for complex equations" but I can't find the sigma-symbol.
Hi!
I have a question regarding my solution to a problem in topology.
Problem: Show that if U is an open connected subset of ℝ2, then U is also path-connected.
Hint: Show that given any x0 in U, show that the set of points that can be joined to x0 by a path in U is both open and closed...