Accumulation points of $A\subseteq C([0,1])$

Question

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

In $(C([0,1]), ||\cdot||_1)$ determine whether the follow are accumulation points of the set A

1) $g_1(t)=0$

2) $g_2(t)=t$

The definition I have for accumulation points is:

Let $(x,d)$ be a metric space and $(x_n)$ a sequence in X. x$\in$X is an accumulation point of the sequence $(x_n)$ if for all $\epsilon >0$ the ball $B(x,e)$ contains $x_n$ for infinitely many n. in this case, we say $(x_n)$ accumulates at x. Let $A \subseteq X, $ Then x$\in$ X is an accumulation point of the set A. If there exists a sequence in A\ {x} that accumulates at x...

I really don't know where I go with this to answer this question

Answer 1

For $j=1,2$, the question is asking you to determine whether there is a sequence of functions $f_n\in C([0,1])$ with $f_n(0)=f_n(1)=0$ and $f_n\neq g_j$ for all $n$, such that $f_n\to g_j$ with respect to the $\lVert\cdot\rVert_1$-norm.

There are easy example sequences in one case, and slightly less easy examples in the other. Try piecewise linear functions for both.

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Expanded Answer: Consider the following functions for all positive integers $n$:

$$f_n(t)=\begin{cases}\frac2nt & 0\le t\le\frac12\\-\frac2n(t-1) & \frac12<t\le1\end{cases}$$ $$h_n(t)=\begin{cases}t & 0\le t\le\frac{n-1}n\\-(n-1)(t-1) & \frac{n-1}n<t\le1\end{cases}$$

You should be able to show that each $f_n$ and each $h_n$ is an element of $A$. Also, you should be able to show that each $f_n\ne g_1$ and each $h_n\ne g_2$. Finally, you should be able to show that the $f_n$ are pairwise distinct, as are the $h_n$. Now, note that for each $n$, $\lVert f_n-g_1\rVert_1$ is simply the area of a triangle with base $1$ and altitude $\frac1n$ (why?) and the same holds for $\lVert h_n-g_2\rVert_1$ (why?). In other words, $$\lVert f_n-g_1\rVert_1=\lVert h_n-g_2\rVert_1=\frac1{2n},$$ so it should be straightforward to show that $f_n\to g_1$ and $h_n\to g_2$ with respect to the $\lVert\cdot\rVert_1$-norm.

Answer 2

Hint: The functions will be accumulation points of $A$ iff you can find a sequence of functions in $A$ (not equal to the limit) that converge to $g_1$ and $g_2$.

To show that a function is an accumulation point, you need to try and find a sequence of functions in $A$, different to the function in question that will converge to that function. To do this, try and find a sequence of functions that "look a lot like" the one you're trying to show is an accumulation point, and then show that the sequence does converge in the required sense.

To show that a function is not an accumulation point, it is often best to assume you have a sequence $a_n \in A$ that converges to the given function and derive a contradiction.