Thread: Show that lim_ x->a+ exists and is finite.

1. Show that lim_ x->a+ exists and is finite.

Suppose that f is uniformly continuous on a bounded half-open interval (a,b]. Show that lim_x-> a+ f(x) exists and is finite. Is the same conclusion true for every bounded continuous function on (a,b]? (If yes show a proof, if no show a counter example).

For this should I use the cauchy criterion theorem which says suppose x_0 in R and f is defined for all x is some deleted neighborhood of x_0 . Suppose for all epsilon >0 there exist delta>0 such that if x,y in N and if 0<|x-x_0|<delta and 0<|y-x_0|<delta then |f(x)-f(y)|<epsilon. then lim_x->x_0 f(x) exists and is finite.

How would i actually use that?

2. Originally Posted by tn11631
Suppose that f is uniformly continuous on a bounded half-open interval (a,b]. Show that lim_x-> a+ f(x) exists and is finite. Is the same conclusion true for every bounded continuous function on (a,b]? (If yes show a proof, if no show a counter example).

For this should I use the cauchy criterion theorem which says suppose x_0 in R and f is defined for all x is some deleted neighborhood of x_0 . Suppose for all epsilon >0 there exist delta>0 such that if x,y in N and if 0<|x-x_0|<delta and 0<|y-x_0|<delta then |f(x)-f(y)|<epsilon. then lim_x->x_0 f(x) exists and is finite.

How would i actually use that?
Let's see some work on the first part!

Spoiler:
$f0,1]\to\mathbb{R}" alt="f0,1]\to\mathbb{R}" /> $x\mapsto\sin\left(\frac{1}{x}\right)$

3. well so far all i pretty much have is that f is defined in the neighborhood (a,a+epsilon)..lol I thought I have to follow the C.criterion. Sorry my brain freezes with proofs...

4. Originally Posted by tn11631
well so far all i pretty much have is that f is defined in the neighborhood (a,a+epsilon)..lol I thought I have to follow the C.criterion. Sorry my brain freezes with proofs...
Write something out. Sometimes I don't know what the hell I'm going to do but when I start writing it comes to me. Try something, I've answered a lot of questions for you. I'd love to see you answer one of your own questions.

5. Originally Posted by Drexel28
Write something out. Sometimes I don't know what the hell I'm going to do but when I start writing it comes to me. Try something, I've answered a lot of questions for you. I'd love to see you answer one of your own questions.
Sorry about all the questions . We get assigned ALOT of hw especially for such a topic. Since the hw takes me like 5 hours I usually ask the ones that I can't think of how to start. But I'll work on it lol thank you

6. Originally Posted by tn11631
Sorry about all the questions . We get assigned ALOT of hw especially for such a topic. Since the hw takes me like 5 hours I usually ask the ones that I can't think of how to start. But I'll work on it lol thank you