## swissgirl Group Title @KingGeorge one year ago one year ago

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1. swissgirl Group Title

2. swissgirl Group Title

Its my weekly question :)

3. swissgirl Group Title

If you dont like this one I can give you another one

4. KingGeorge Group Title

It's been a little while since I've done actual analysis :P

5. swissgirl Group Title

Ya so let me get you smth easier

6. KingGeorge Group Title

Well, the first part is pretty easy. It's basically just the definition of "not uniformly continuous"

7. swissgirl Group Title

BTW Congratulations. Where is ur offer from?

8. swissgirl Group Title

9. KingGeorge Group Title

U of Rochester. I'm still waiting to hear back from most schools I applied to. Unfortunately, I also just recently got a rejection from Princeton :(

10. swissgirl Group Title

awwwwww :( That is sad They are stupiidddddd

11. KingGeorge Group Title

Ah. This new one I can do. It's just the IVP (intermediate value theorem). To be honest, I wasn't expecting them to accept me.

12. KingGeorge Group Title

Let's see how much I remember. Last time proved this, we started with the assumption that $$f(a)f(b)\le 0$$. In this particular case, if $$f(a)=0$$ or $$f(b)=0$$, we're done. If $$f(a)f(b)<0$$, then one of $$f(a),f(b)$$ is negative, and the other positive. At this point, we had another theorem that proved there was then some $$c\in[a,b]$$ s.t. $$f(c)=0$$. Then define $$g(x)=f(x)+y$$. Then, from what we just proved, it immediately follows that $$g(c)=y$$. If we started with $$g(b)\le y\le g(a)$$, take $$f(x)=g(x)-y$$, so $$f(b)\le 0 \le g(a)$$. From this we get that $$f(b)f(a)\le 0$$ and we're done.

13. KingGeorge Group Title

However, I cannot immediately remember how we proved that if $$f(a)f(b)\le0$$, then there was some $$c\in[a,b]$$ s.t. $$f(c)=0$$.

14. swissgirl Group Title

hmmm gonna check my notes

15. swissgirl Group Title

hmmm I found a proof in my book using the supremum

16. KingGeorge Group Title

Well, I'm sorry, but I've got to go now. I need to get up early tomorrow, so I need to head to bed. Have a good night, and good luck with this!

17. swissgirl Group Title

Thanks @KG :)

18. swissgirl Group Title

Good night