Ir al contenido principal

## Due September 11th

### Problem 1

Let $f:X\to Y$ be a function, $A,B\subset X$ and $U,V\subset Y$.

1. $f(A\cup B) = f(A)\cup f(B)$.

2. $f(A\cap B)\subset f(A)\cap f(B)$. Give an example where $f(A\cap B)\not\supset f(A)\cap f(B)$.

3. $f^{-1}(U\cup V) = f^{-1}(U)\cup f^{-1}(V)$.

4. $f^{-1}(U\cap V) = f^{-1}(U)\cap f^{-1}(V)$.

5. $f(f^{-1}(U)) \subset U$. Give an example where $f(f^{-1}(U))\not\supset U$.

6. $f^{-1}(f(A)) \supset A$. Give an example where $f^{-1}(f(A))\not\subset A$.

### Problem 2

If $X$ is sequentially compact and $f:X\to Y$ is continuous, then $f(X)$ is sequentially compact. Prove it directly using the definition of sequential compactness.

### Problem 3

Give a set $X$ and two metrics $d,d'$ on $X$ such that $(X,d)$ and $(X,d')$ are homeomorphic, but $f:X\to X$ given by $f(x)=x$ is not uniformly continuous.

### Problem 4

1. Let $\mathcal I:C([0,1]\to C([0,1])$ be the operator given by

$\displaystyle \mathcal If(x) = \int_0^x f(t) dt,$

i.e. $\mathcal If$ is the undefined integral of $f$. Then $\mathcal I$ is continuous with respect to the uniform norm.

2. Use (1) to prove the following theorem:
Let $f_n\in C^1([0,1])$ and $g\in C([0,1])$ such that

1. $(f_n(x_0))$ converges for some $x_0$

2. $f_n' \rightrightarrows g$

Then $f_n$ converges uniformly and, if $f_n\rightrightarrows f$, then $f\in C^1([0,1])$ and $f'=g$.

### Problem 5

1. Let $C_0(X,\R)$ be the space of continuous functions $f:X\to\R$ that go to 0 at infinity, i. e. for every $\e>0$ there exists a compact $E\subset X$ such that $|f(x)|<\e$ for every $x\not\in X$.
Then $C_0(X,\R)$ is a closed subspace of $C_B(X,\R)$.

2. Let $C_c(X,\R)$ be the space of continuous functions $f:X\to\R$ of compact support, i. e. the closure of the set $\{x\in X:f(x)\not=0\}$ is compact.
Then $C_c(X,\R)$ is a subspace of $C_B(X,\R)$. Give an example where it is not closed in $C_c(X,\R)$.