Tarea 5, Análisis real

Due September 11th

Problem 1

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

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


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


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


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


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


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

Problem 2

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

Problem 3

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

Problem 4

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

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

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

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

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


2. $latex f_n' \rightrightarrows g$
   

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

Problem 5

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


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