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1. a) Let $n,m$ be integers $\geq 1$. Find a discontinuous function $f$ from $\mathbb R^n$ to $\mathbb R^m$ such that $f$ sends compact sets to compact sets. b) Find $n$, $m$ and $f$ a discontinuous function from $\mathbb R^n$ to $\mathbb R^m$ that sends connected sets to connected sets c) Let $f$ a function from $\mathbb R^n$ to $\mathbb R^m$ that sends connected sets to connected sets and compact sets to compact sets. Prove that $f$ is continuous 2. Find all continous group homorphisms from $SO_2(\mathbb R)$ to $GL_n(\mathbb R)$ 3. Let $A$ a square complex matrix that has $0$ trace. Prove $A$ is similar to a matrix with only $0$ on its diagonal. 4. Let $a$ be an odd integer. Does there exist a bijection in $\mathbb Z$ such that $\forall k\in \mathbb Z, f(f(k))=a+k$ ? 5. Let $u_n$ be a real sequence such that $u_{n+1}-u_n-u_n^2\to 0$. Show that $u$ goes either to $0$ or to $\infty$
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Tue, 17 Jun 2014 14:02 GMT