Prove or disprove each of the statements below.
(A) (4 marks) Let $f : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R }$ be a continuous function that takes both positive and negative values. Then $f$ has infinitely many zeros.
(B) (6 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$ be a continuous function. Then $f$ is not open.

It is known that there exist surjective continuous maps $I \longrightarrow I ^ { 2 }$ where $I = [ 0,1 ]$ is the unit interval.
(A) (4 marks) Using the above result or otherwise, show that there exists a surjective continuous map $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$.
(B) (6 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$ be a surjective continuous map. Let $\Gamma = \{ ( x , f ( x ) ) \mid x \in \mathbb { R } \} \subset \mathbb { R } ^ { 3 }$. Show that $\mathbb { R } ^ { 3 } \setminus \Gamma$ is path connected.

We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$.
a) Prove that $K$ is a compact subset of $\mathbf{R}^3$ for its usual topology.
b) Prove that $K$ is convex, that is, for every real $t\in[0,1]$ and every pair $(u,v)$ of elements of $K$, $tu+(1-t)v$ belongs to $K$.
c) Establish that, if $(a,b,c)\in\mathbf{C}^3$, $\widehat{abc}$ is a compact convex subset of $\mathbf{C}$ equipped with its usual topology.
d) With the same notation prove the existence of: $$\delta(\widehat{abc}) = \max\left\{|z'-z| / (z,z')\in\widehat{abc}^2\right\}$$

We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$.
a) Prove that, if we fix $z\in\mathbf{C}$ and $(a,b,c)\in\mathbf{C}^3$: $$\max\left\{|z'-z| / z'\in\widehat{abc}\right\} = \max(|z-a|,|z-b|,|z-c|)$$
b) Deduce from this a simple expression for $\delta(\widehat{abc})$.

For every real $\alpha > 1$, show that $1 \leqslant S ( \alpha ) \leqslant 1 + \frac { 1 } { \alpha - 1 }$, where $S ( \alpha ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { \alpha } }$.

We define a sequence of polynomials $\left( A _ { n } \right) _ { n \in \mathbb { N } }$ satisfying $A _ { 0 } = 1 , A _ { n + 1 } ^ { \prime } = A _ { n }$ and $\int _ { 0 } ^ { 1 } A _ { n + 1 } ( t ) d t = 0$ for every $n \in \mathbb { N }$. For $n \geqslant 2$, the variations of $A_n$ on $[0,1]$ are known, in particular:

1. If $n \equiv 2 \bmod 4$, then $A _ { n } ( 0 ) = A _ { n } ( 1 ) > 0 > A _ { n } \left( \frac { 1 } { 2 } \right)$.
2. If $n \equiv 0 \bmod 4$, then $A _ { n } ( 0 ) = A _ { n } ( 1 ) < 0 < A _ { n } \left( \frac { 1 } { 2 } \right)$.
3. If $n \equiv 1 \bmod 4$, then $A _ { n } ( 0 ) = A _ { n } \left( \frac { 1 } { 2 } \right) = A _ { n } ( 1 ) = 0$; $A _ { n } < 0$ on $]0 , \frac{1}{2}[$ and $A _ { n } > 0$ on $]\frac{1}{2} , 1[$.
4. If $n \equiv 3 \bmod 4$, then $A _ { n } ( 0 ) = A _ { n } \left( \frac { 1 } { 2 } \right) = A _ { n } ( 1 ) = 0$; $A _ { n } > 0$ on $]0 , \frac{1}{2}[$ and $A _ { n } < 0$ on $]\frac{1}{2} , 1[$.

a) Show that for $n \geqslant 2$, the variations of the polynomials $A _ { n }$ on $[ 0,1 ]$ correspond to the four cases described above.
b) For every $n \in \mathbb { N } ^ { * }$ and every $x \in [ 0,1 ]$, show that $\left| A _ { 2 n } ( x ) \right| \leqslant \left| a _ { 2 n } \right|$ and $\left| A _ { 2 n + 1 } ( x ) \right| \leqslant \frac { \left| a _ { 2 n } \right| } { 2 }$.

Let $n \in \mathbb { N }$ and let $f$ be a real function of class $C ^ { \infty }$ on $[ n , + \infty [$. We assume that $f$ and all its derivatives have constant sign on $[ n , + \infty [$ and tend to 0 as $+ \infty$.
By applying, for $k \geqslant n$, the result of III.B.1 to $f _ { k } ( t ) = f ( k + t )$, show $$\sum _ { k = n } ^ { + \infty } f ^ { \prime } ( k ) = - f ( n ) + \frac { 1 } { 2 } f ^ { \prime } ( n ) - \sum _ { j = 1 } ^ { p } a _ { 2 j } f ^ { ( 2 j ) } ( n ) + \int _ { n } ^ { + \infty } A _ { 2 p + 1 } ^ { * } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t$$ where we have set $A _ { j } ^ { * } ( t ) = A _ { j } ( t - [ t ] )$ for every $t \in \mathbb { R }$.
Show that $$\left| \int _ { n } ^ { + \infty } A _ { 2 p + 1 } ^ { * } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t \right| \leqslant \left| \frac { a _ { 2 p } } { 2 } \right| \left| f ^ { ( 2 p + 1 ) } ( n ) \right|$$

Let $g$ be a piecewise continuous increasing function on $[ 0,1 ]$.
By noting $\int _ { 0 } ^ { 1 } = \int _ { 0 } ^ { 1 / 2 } + \int _ { 1 / 2 } ^ { 1 }$, show that

• if $n \equiv 1 \bmod 4$, then $\int _ { 0 } ^ { 1 } A _ { n } ( t ) g ( t ) d t \geqslant 0$;
• if $n \equiv 3 \bmod 4$, then $\int _ { 0 } ^ { 1 } A _ { n } ( t ) g ( t ) d t \leqslant 0$.

Using the notation from II.B.2, where $\widetilde { S } _ { n , 2 p - 2 } ( \alpha ) = \sum _ { k = 1 } ^ { n - 1 } \frac { 1 } { k ^ { \alpha } } - \left( a _ { 0 } f ( n ) + a _ { 1 } f ^ { \prime } ( n ) + \cdots + a _ { 2 p - 2 } f ^ { ( 2 p - 2 ) } ( n ) \right)$, show that for every natural integer $p \geqslant 1$ $$\widetilde { S } _ { n , 4 p } ( \alpha ) \leqslant S ( \alpha ) \leqslant \widetilde { S } _ { n , 4 p + 2 } ( \alpha )$$ and that $$\widetilde { S } _ { n , 4 p } ( \alpha ) \leqslant S ( \alpha ) \leqslant \widetilde { S } _ { n , 4 p - 2 } ( \alpha )$$ Deduce that the error $\left| S ( \alpha ) - \widetilde { S } _ { n , 2 p } ( \alpha ) \right|$ is bounded by $\left| a _ { 2 p + 2 } f ^ { ( 2 p + 2 ) } ( n ) \right|$.

Show that, for all integers $n , p \geqslant 1$ $$\left| \frac { a _ { 2 p + 2 } f ^ { ( 2 p + 2 ) } ( n ) } { a _ { 2 p } f ^ { ( 2 p ) } ( n ) } \right| = \frac { ( \alpha + 2 p ) ( \alpha + 2 p - 1 ) S ( 2 p + 2 ) } { 4 n ^ { 2 } \pi ^ { 2 } S ( 2 p ) }$$ where $f$ is defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$.

In each of the two cases below, show that $f * g$ is defined and bounded on $\mathbb{R}$ and give an upper bound for $\|f * g\|_{\infty}$ which may involve $\|\cdot\|_{1}$, $\|\cdot\|_{2}$ or $\|\cdot\|_{\infty}$. a) $f \in L^{1}(\mathbb{R}),\ g \in C_{b}(\mathbb{R})$; b) $f, g \in L^{2}(\mathbb{R})$.

For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. For any real $\alpha$, show that $\left\|T_{\alpha}(f * g) - f * g\right\|_{\infty} \leqslant \left\|T_{\alpha}(f) - f\right\|_{2} \times \|g\|_{2}$.

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $A _ { n } = \sum _ { k = 0 } ^ { n } a _ { k }$.
For all $x \in \left[ 0,1 \left[ \right. \right.$ and all $n \in \mathbb { N }$, show that $f ( x ) \geqslant A _ { n } x ^ { n }$.

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$.
Show the existence of an integer $N > 0$ such that $$\forall n \geqslant N , \quad f \left( \mathrm { e } ^ { - 1 / n } \right) \leqslant \frac { 2 } { 1 - \mathrm { e } ^ { - 1 / n } }$$

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $A _ { n } = \sum _ { k = 0 } ^ { n } a _ { k }$ and $\widetilde { a } _ { n } = \frac { A _ { n } } { n + 1 }$.
Deduce that the sequence $\left( \widetilde { a } _ { n } \right) _ { n \geqslant 0 }$ is bounded above.

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $A _ { n } = \sum _ { k = 0 } ^ { n } a _ { k }$, $\widetilde { a } _ { n } = \frac { A _ { n } } { n + 1 }$, and $\mu > 0$ is an upper bound of $\left( \widetilde { a } _ { n } \right)$.
Let $\lambda$ be a strictly positive real number. a) Show that there exists an integer $N _ { 0 } > 0$ such that for all $N \geqslant N _ { 0 }$, $$f \left( \mathrm { e } ^ { - \lambda / N } \right) \geqslant \frac { 1 } { 2 \left( 1 - \mathrm { e } ^ { - \lambda / N } \right) } \geqslant \frac { N } { 2 \lambda } .$$ b) Show that for all $N \geqslant N _ { 0 }$ $$\tilde { a } _ { N - 1 } \geqslant \frac { 1 } { 2 \lambda } - \mu \mathrm { e } ^ { - \lambda } \left( 1 + \frac { 1 } { N } + \mathrm { e } ^ { - \lambda / N } \frac { 1 } { N \left( 1 - \mathrm { e } ^ { - \lambda / N } \right) } \right)$$ c) Determine as a function of $\lambda$ the limit, when $N$ tends to infinity, of the right-hand side in the previous inequality. d) Show that there exists a real $\lambda > 0$ such that this limit is strictly positive.

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. Let $g$, $g^+$, $g^-$, $P$, $Q$ be as defined in II.E.1--II.E.3. For every integer $N > 0$ we set $x _ { N } = \mathrm { e } ^ { - 1 / N }$.
Establish the existence of an integer $N _ { 1 } > 0$ such that for every integer $N \geqslant N _ { 1 }$, $$\left( 1 - x _ { N } \right) \sum _ { n = 0 } ^ { + \infty } a _ { n } x _ { N } ^ { n } P \left( x _ { N } ^ { n } \right) \geqslant \int _ { 0 } ^ { 1 } P ( t ) \mathrm { d } t - \varepsilon$$ and $$\left( 1 - x _ { N } \right) \sum _ { n = 0 } ^ { + \infty } a _ { n } x _ { N } ^ { n } Q \left( x _ { N } ^ { n } \right) \leqslant \int _ { 0 } ^ { 1 } Q ( t ) \mathrm { d } t + \varepsilon$$

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $A_N = \sum_{k=0}^N a_k$. For every integer $N > 0$ we set $x _ { N } = \mathrm { e } ^ { - 1 / N }$. Let $N_1$ be as in II.E.4.
Deduce from the three previous questions that for every integer $N \geqslant N _ { 1 }$ $$1 - 5 \varepsilon \leqslant \left( 1 - x _ { N } \right) A _ { N } \leqslant 1 + 5 \varepsilon$$

Justify that for real $x$, $\left| \varphi _ { n } ( x ) \right| \leqslant 1$.

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be a real decreasing sequence converging to 0, and $\left( b _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be a complex sequence such that the sequence $\left( B _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ defined for all $n \in \mathbb { N } ^ { * }$ by $B _ { n } = b _ { 1 } + \cdots + b _ { n }$ is bounded.
Show that, for all integer $n \geqslant 2$, $$\sum _ { k = 1 } ^ { n } a _ { k } b _ { k } = a _ { n } B _ { n } + \sum _ { k = 1 } ^ { n - 1 } \left( a _ { k } - a _ { k + 1 } \right) B _ { k }$$

We consider the function series $\sum _ { n \geq 1 } \frac { \sin ( n x ) } { \sqrt { n } }$, where $x$ is a real variable.
Show that it cannot be the Fourier series of a $2 \pi$-periodic piecewise continuous function.
One may begin by recalling Parseval's formula.

For all $f \in \mathcal{C}$, the function $\omega_{f} : [0,1] \rightarrow \mathbf{R}_{+}$ is defined by $$\omega_{f}(h) = \sup \{|f(x) - f(y)| \mid x, y \in [0,1] \text{ and } |x - y| \leq h\} .$$ Show that $\omega_{f}$ is increasing, and continuous at 0.

For all $f \in \mathcal{C}$, the function $\omega_{f} : [0,1] \rightarrow \mathbf{R}_{+}$ is defined by $$\omega_{f}(h) = \sup \{|f(x) - f(y)| \mid x, y \in [0,1] \text{ and } |x - y| \leq h\} .$$ Show that for all $h, h' \in [0,1]$ such that $h \leq h'$, $\omega_{f}$ satisfies $$\omega_{f}(h') \leq \omega_{f}(h) + \omega_{f}(h' - h) .$$

Let $s \in [0,1[$. Suppose that the function $h \mapsto \frac{\omega_{f}(h)}{h^{s}}$ is bounded on $]0,1]$. For all $x_{0} \in [0,1]$, show that $f \in \Gamma^{s}(x_{0})$.

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. We recall that $\widetilde{k}_{j}(x)$ is the integer part of $2^{j} x$. We set $$W_{j} = \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x) - \theta_{j,k}(x_{0})|$$ Show that $$W_{j} \leq (|c_{j,\widetilde{k}_{j}(x)}(f)| + |c_{j,\widetilde{k}_{j}(x_{0})}(f)|) 2^{j+1} |x - x_{0}|$$