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## Due September 25th

### Problem 1

Let $f,g\in C(X)$. Then the functions

$\max(f,g):X\to\R$ and $\min(f,g):X\to\R$

are continuous.

### Problem 2

Let $a>0$. The space of continous even functions on $[-a,a]$ is a proper closed subalgebra of $C([-a,a])$.

### Problem 3

Let $f$ be a continuous function on $[a,b]$ such that

$\displaystyle \int_a^b f(x) x^n dx = 0$

for all $n=0,1,2,\ldots$. Then $f=0$.

### Problem 4

Find a nontrivial continuous function $f:[0,1]\to\R$ such that

$\displaystyle \int_0^1 f(x)dx = \int_0^1 f(x) x dx = \int_0^1 f(x)x^2dx = 0.$

### Problem 5

Let $X,Y$ compact metric spaces, $X\times Y$ the product space and $\mathcal A$ the algebra generated by the functions $f:X\times Y\to \R$ of the form

$f(x,y) = g(x)h(y), \qquad g\in C(X), \; h\in C(Y).$

Then $\mathcal A$ is dense in $C(X\times Y)$.