I have a few practice problems attached in a .pdf. Help please?

2. Where are you stuck?

Note that the first question is false as stated: Consider $\displaystyle \mathbb{S} ^1$ as a metric subspace of $\displaystyle \mathbb{R} ^2$ and let $\displaystyle f$ be a rotation about the origin of $\displaystyle \pi$ then $\displaystyle f$ has no fixed points on $\displaystyle \mathbb{S} ^1$ but $\displaystyle f^2$ is the identity.

3. The question is actually false?? :S

I'm kind of stuck on a lot of it... but questions 1, 2 and 5 are the hardest at the moment for me. Would you be able to help me get started on them?

4. Originally Posted by Shajialin
The question is actually false?? :S

I'm kind of stuck on a lot of it... but questions 1, 2 and 5 are the hardest at the moment for me. Would you be able to help me get started on them?
Question one, as I showed you is false as it was stated, you need to ask the fixed point be unique.

For question two, notice that a strictly increasing function always has a left and right limit (even if they do not coincide) so the only discontinuities it can have are jumps, but the image is an interval.

For five, you need that composition of unif. cont. functions is again unif. cont. and the following fact: If $\displaystyle f:T\rightarrow U$ is continous and bijective and $\displaystyle T$ is compact then $\displaystyle f^{-1}$ is continous (to prove this use sequential compactness of $\displaystyle T$) so $\displaystyle g=f^{-1} \circ (f \circ g)$ and you're done.

5. Thank you for your help so far! Just to check what you would do though... how is it that you would approach questions 3 and 4?

6. For three pick $\displaystyle n$ big enough such that $\displaystyle \frac{1}{n} \leq \epsilon$ then the rationals for which $\displaystyle \vert g(x) \vert \geq \epsilon \geq \frac{1}{n}$ are finite.

For 4 just use the triangle inequality to get that if $\displaystyle f_n \rightarrow f$ then $\displaystyle \vert f(x) \vert \leq \vert f_n(x) \vert + \epsilon$ so for $\displaystyle \vert x \vert$ big enough and $\displaystyle \epsilon$ small enough $\displaystyle \vert f(x) \vert$ is also small.

7. Thank you very much for your input! I'm still trying to prove the first one, but it is starting to make sense. Thanks again!

8. ## One more question...

While you are still online, could I get your input on the following question, out of interest?

Let f be strictly increasing on a subset S of R. Assume that the image f(S) has one of the following properties:
(a) f(S) is open
(b) f(S) is connected
(c) f(S) is closed

Prove that f must be continuous on S.

9. If it's connected it's an interval. For the other two assume there is a discontinuity and derive that one of the components of f(S) has an endpoint while another one doesn't.

10. That makes sense for when it is connected. I'll work on open/ closed for a bit more.