Real Analysis: continuity

Quote:

Originally Posted by Vitava61

Sorry, only a few more, lasts ones I'm not sure of, final on Friday, these are some of the review problems I don't get:

1) Show 2^x = 3x for some xE (0,1)

2) let f(x) = { x xrational
0 xirrational

a. prove lim f(x) = 0
x->0
b) Prove that lim f(x), does not exist for c not equal to 0.
x--> c

3) Suppose f is uniformly continuous on [a,b] and uniformly continuous on [b,c] Prove using an epsilon-delta argument that f is uniformly continuous on [a,c]

Thanks, this professor does a horrible job at teaching, and I'm really lost, any help, even how to start any of these would be great, thanks!

for 1)
consider the function $\displaystyle f(x)=2^x-3x$
Hint: use the Intermediate Value Threom

for 2) How would you prove

$\displaystyle \lim_{x \to 0}f(x)=0$ if $\displaystyle f(x)=x$

compare with the above problem and adapt the proof.

for 3)
uniform continity state that for every epslion > 0 there exits a delta

fix epsilon > 0 find a delta on [a,b] and [b,c]

How could we get a delta that would work on [a,c]?

I hope this helps.

Brett