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# Accumulation Points watch

1. Consider the subset A of C([0,1]) consisting of continuous function f with f(0)=f(1)=0

In determine whether the following are accumulation points of the set A

1)
2)

How would i go about solving this type of question?
2. (Original post by gemma331)
Consider the subset A of C([0,1]) consisting of continuous function f with f(0)=f(1)=0

In determine whether the following are accumulation points of the set A

1)
2)

How would i go about solving this type of question?
Do you know the definition of an accumulation point? Which of g_1 and g_2 intuitively seems like it could be an accumulation point?
3. (Original post by Mark13)
Do you know the definition of an accumulation point? Which of g_1 and g_2 intuitively seems like it could be an accumulation point?

The one i think is more likely to be an accumulation point is g_1 but im really not sure how i go about solving this... the examples i have previously done.. in previous work i have been given an and find the accumulation points through where

Im not really sure where i go with this question, but proving which one is and proving why the other isnt.
4. (Original post by gemma331)
The one i think is more likely to be an accumulation point is g_1 but im really not sure how i go about solving this... the examples i have previously done.. in previous work i have been given an and find the accumulation points through where

Im not really sure where i go with this question, but proving which one is and proving why the other isnt.
The first thing you need to do is go back to your lecture notes/text book and get familiar with what an accumulation point is - feel free to post any questions you have about the definition up here.
5. (Original post by Mark13)
The first thing you need to do is go back to your lecture notes/text book and get familiar with what an accumulation point is - feel free to post any questions you have about the definition up here.

I know I have to find two sequences of functions in A one t hat converges to and the other to but I really don't know how I go about doing ththis please help!!
6. (Original post by gemma331)
I know I have to find two sequences of functions in A one t hat converges to and the other to but I really don't know how I go about doing ththis please help!!
Think about an arbitrary function f in A. Then we know that f is continuous, and f(0)=f(1)=0. Also, .

Can you see how to use this to show that g_2 isn't an accumulation point of A?

g_1 is an accumulation point of A - to show this you need to find a sequence of continuous functions, each of which takes the value 0 at x=0 and x=1, which converge to g_1 in the uniform norm.
7. (Original post by Mark13)
Think about an arbitrary function f in A. Then we know that f is continuous, and f(0)=f(1)=0. Also, .

Can you see how to use this to show that g_2 isn't an accumulation point of A?
I still don't understand how this shows g_2 isn't an accumulation point :s
8. (Original post by gemma331)
I still don't understand how this shows g_2 isn't an accumulation point :s
If g_2 is an accumulation point, then there is a sequence of functions f_n in A such that . If you can show that for any function f in A, we have , then no such sequence f_n can exist, so g_2 is not an accumulation point.
9. (Original post by Mark13)
...
Just noting that you seem to have switched from the 1-norm to the infinity-norm.
10. (Original post by ghostwalker)
Just noting that you seem to have switched from the 1-norm to the infinity-norm.
Thanks for pointing that out, I need to read things more carefully

In that case, both functions are accumulation points, and to show that a given function g is an accumulation point, you need to find a sequence of functions f_n such that .

Often a good way to find a sequence of functions converging to the required limit is to think graphically. For example, the sequence of functions f_n defined by:

f_n agrees with g_2 for all x satisfying , and then linearly intepolating between the points (1-1/n,1-1/n) and (1,0) after that (it's a lot easier to draw this function than give an expression for it)

will converge to g_2 in the 1-norm.

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Updated: March 28, 2013
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