For $n \in \mathbb { Z }$, we denote by $\mathcal { E } _ { n }$ the space of functions $f$ in $\mathcal { C } ^ { 2 } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { C } \right)$ such that $$\forall t \in \mathbb { R } _ { + } ^ { * } , \quad t ^ { 2 } f ^ { \prime \prime } ( t ) + t f ^ { \prime } ( t ) - n ^ { 2 } f ( t ) = 0$$ For $\alpha \in \mathbb { R }$, let $\varphi _ { \alpha }$ be the function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ defined by $$\forall t \in \mathbb { R } _ { + } ^ { * } , \quad \varphi _ { \alpha } ( t ) = t ^ { \alpha }$$
For all $n \in \mathbb { Z } ^ { * }$, determine the real numbers $\alpha$ such that $\varphi _ { \alpha }$ belongs to $\mathcal { E } _ { n }$.

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ and $\widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. For $n \in \mathbb { Z }$, let $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad c _ { n , f } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { f } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta, \quad c _ { n , g } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { g } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta$$
Show that $c _ { n , f }$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$ and satisfies $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad \left( c _ { n , f } \right) ^ { \prime } ( r ) = \frac { i n } { r } c _ { n , g } ( r )$$

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ and $\widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. For $n \in \mathbb { Z }$, let $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad c _ { n , f } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { f } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta, \quad c _ { n , g } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { g } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta$$ For $n \in \mathbb { Z }$, $\mathcal { E } _ { n }$ denotes the space of functions $f$ in $\mathcal { C } ^ { 2 } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { C } \right)$ such that $t ^ { 2 } f ^ { \prime \prime } ( t ) + t f ^ { \prime } ( t ) - n ^ { 2 } f ( t ) = 0$ for all $t \in \mathbb { R } _ { + } ^ { * }$.
Show that $c _ { n , f }$ belongs to $\mathcal { E } _ { n }$ and that $c _ { n , f }$ is bounded in a neighbourhood of 0. Deduce the existence of $a _ { n } \in \mathbb { C }$ such that $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad c _ { n , f } ( r ) = a _ { n } r ^ { | n | }$$

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. From the previous questions, there exist $a_n \in \mathbb{C}$ such that $c_{n,f}(r) = a_n r^{|n|}$ for all $r \in \mathbb{R}_+^*$.
By stating precisely the theorem used, establish $$\forall ( r , \theta ) \in \mathbb { R } _ { + } ^ { * } \times \mathbb { R } , \quad \widetilde { f } ( r , \theta ) = \lim _ { p \rightarrow \infty } \sum _ { n = - p } ^ { p } a _ { n } r ^ { | n | } e ^ { i n \theta }$$

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ and $\widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. For $n \in \mathbb { Z }$, let $c _ { n , f } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { f } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta$. We assume that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are bounded on $\mathbb { R } ^ { 2 }$.
If $n \in \mathbb { Z }$, show that the function $\left( c _ { n , f } \right) ^ { \prime }$ is bounded on $\mathbb { R } _ { + } ^ { * }$.

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We assume that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are bounded on $\mathbb { R } ^ { 2 }$.
Show that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are constant.

We denote by $\mathcal { P } _ { 2 }$ the set of polynomial functions of degree $\leqslant 2$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$, that is, the maps from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$ of the form $$( x , y ) \mapsto a x ^ { 2 } + b x y + c y ^ { 2 } + d x + e y + f \quad \text { where } \quad ( a , b , c , d , e , f ) \in \mathbb { R } ^ { 6 }$$ A function $f \in \mathcal{C}^2(\mathbb{R}^2, \mathbb{R})$ satisfies (1) if and only if $$\forall ( x , y ) \in \mathbb{R}^2, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$
Determine the functions in $\mathcal { P } _ { 2 }$ satisfying (1) on $\mathbb { R } ^ { 2 }$.

If $I$ is an interval of $\mathbb { R }$, we say that $u \in \mathcal { C } ^ { 1 } ( I , \mathbb { R } )$ satisfies (II.1) on $I$ if and only if $$\forall t \in I , \quad u ( t ) \left( u ( t ) + 2 t u ^ { \prime } ( t ) \right) = - 1$$ A function $f \in \mathcal{C}^2(\mathbb{R}^2, \mathbb{R})$ satisfies (1) if and only if $$\forall ( x , y ) \in \mathbb{R}^2, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ Let $J$ be a non-empty open interval of $\mathbb { R }$, $\Omega ( J ) = \left\{ ( x , y ) \in \mathbb { R } ^ { 2 } , x y \in J \right\}$, $w$ in $\mathcal { C } ^ { 2 } ( J , \mathbb { R } )$ and $W$ the function defined by $$\forall ( x , y ) \in \Omega ( J ) , \quad W ( x , y ) = w ( x y )$$

1. [II.D.1)] Show that $\Omega ( J )$ is a non-empty open set.
2. [II.D.2)] Show that $W$ is in $\mathcal { C } ^ { 2 } ( \Omega ( J ) , \mathbb { R } )$ and that there is equivalence between
   1. [i.] $W$ satisfies (1) on $\Omega ( J )$,
   2. [ii.] $w ^ { \prime }$ satisfies (II.1) on $J$.
3. [II.D.3)] Show that $W$ is the restriction to $\Omega ( J )$ of a function in $\mathcal { P } _ { 2 }$ if and only if $w$ is affine.

A function $f \in \mathcal{C}^2(\Omega, \mathbb{R})$ satisfies (1) on $\Omega$ if and only if $$\forall ( x , y ) \in \Omega, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ Let $\Omega$ be a non-empty open set of $\mathbb { R } ^ { 2 }$, $f$ in $\mathcal { C } ^ { 2 } ( \Omega , \mathbb { R } )$ satisfying (1) on $\Omega$, $( a , b ) \in \mathbb { R } ^ { 2 }$, $\Omega _ { a , b }$ the image of $\Omega$ by the translation of vector $( a , b )$ and $f _ { a , b }$ the function defined on $\Omega _ { a , b }$ by $$\forall ( x , y ) \in \Omega _ { a , b } , \quad f _ { a , b } ( x , y ) = f ( x - a , y - b )$$
Show that $f _ { a , b }$ satisfies (1) on $\Omega _ { a , b }$.

A function $f \in \mathcal{C}^2(\mathbb{R}^2, \mathbb{R})$ satisfies (1) on $\mathbb{R}^2$ if and only if $$\forall ( x , y ) \in \mathbb{R}^2, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ We denote by $\mathcal { P } _ { 2 }$ the set of polynomial functions of degree $\leqslant 2$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$.
If $( x _ { 0 } , y _ { 0 } )$ is in $\mathbb { R } ^ { 2 }$, show that there exists an open set $U$ of $\mathbb { R } ^ { 2 }$ containing $( x _ { 0 } , y _ { 0 } )$ such that the set of functions in $\mathcal { C } ^ { 2 } ( U , \mathbb { R } )$ satisfying (1) on $U$ and not coinciding on $U$ with any element of $\mathcal { P } _ { 2 }$ is infinite.

Recall the definition of a $\mathcal { C } ^ { 1 }$-diffeomorphism of $\mathbb { R } ^ { 2 }$ onto $\mathbb { R } ^ { 2 }$ and the theorem characterizing such a diffeomorphism among applications of class $\mathcal { C } ^ { 1 }$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R } ^ { 2 }$.

We consider $\alpha \in \mathbb { R } _ { + } ^ { * }$ and $F \in \mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } ^ { 2 } \right)$. We assume that for all $( p , h ) \in \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 }$ $$\left\langle d F _ { p } ( h ) , h \right\rangle \geqslant \alpha \| h \| ^ { 2 }$$ Let $p$ and $q$ be in $\mathbb { R } ^ { 2 }$.

1. [III.B.1)] Verify $$F ( q ) - F ( p ) = \int _ { 0 } ^ { 1 } d F _ { p + t ( q - p ) } ( q - p ) \mathrm { d } t$$
2. [III.B.2)] Show $$\langle F ( q ) - F ( p ) , q - p \rangle \geqslant \alpha \| q - p \| ^ { 2 }$$

We consider $\alpha \in \mathbb { R } _ { + } ^ { * }$ and $F \in \mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } ^ { 2 } \right)$. We assume that for all $( p , h ) \in \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 }$ $$\left\langle d F _ { p } ( h ) , h \right\rangle \geqslant \alpha \| h \| ^ { 2 }$$ Let $a \in \mathbb { R } ^ { 2 }$ and $G ^ { a }$ be the map from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$ defined by $$\forall p \in \mathbb { R } ^ { 2 } , \quad G ^ { a } ( p ) = \| F ( p ) - a \| ^ { 2 }$$

1. [III.C.1)] If $p$ and $h$ are in $\mathbb { R } ^ { 2 }$, compute $d G ^ { a } { } _ { p } ( h )$.
2. [III.C.2)] Show that $G ^ { a } ( p ) \rightarrow + \infty$ when $\| p \| \rightarrow + \infty$.
3. [III.C.3)] Deduce that $G ^ { a }$ attains a global minimum on $\mathbb { R } ^ { 2 }$ at a point $p _ { 0 }$.
4. [III.C.4)] Show that $F \left( p _ { 0 } \right) = a$.

We consider $\alpha \in \mathbb { R } _ { + } ^ { * }$ and $F \in \mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } ^ { 2 } \right)$. We assume that for all $( p , h ) \in \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 }$ $$\left\langle d F _ { p } ( h ) , h \right\rangle \geqslant \alpha \| h \| ^ { 2 }$$
Show that $F$ realizes a $\mathcal { C } ^ { 1 }$-diffeomorphism of $\mathbb { R } ^ { 2 }$ onto $\mathbb { R } ^ { 2 }$.

Let $f$ be in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$: $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ For $( x , y ) \in \mathbb { R } ^ { 2 }$, let $u ( x , y ) = x + \frac { \partial f } { \partial x } ( x , y ) , v ( x , y ) = y + \frac { \partial f } { \partial y } ( x , y )$ and $F ( x , y ) = ( u ( x , y ) , v ( x , y ) )$. We assume that $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) > 0$ for all $( x , y ) \in \mathbb { R } ^ { 2 }$.
If $( x , y ) \in \mathbb { R } ^ { 2 }$, show that $\operatorname { Jac } F ( x , y ) - I _ { 2 }$ (where $I _ { 2 }$ denotes the identity matrix of order 2) is symmetric positive semidefinite. Deduce that $F$ is a $\mathcal { C } ^ { 1 }$-diffeomorphism of $\mathbb { R } ^ { 2 }$ onto $\mathbb { R } ^ { 2 }$.

Let $f$ be in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$: $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ For $( x , y ) \in \mathbb { R } ^ { 2 }$, let $u ( x , y ) = x + \frac { \partial f } { \partial x } ( x , y ) , v ( x , y ) = y + \frac { \partial f } { \partial y } ( x , y )$ and $F ( x , y ) = ( u ( x , y ) , v ( x , y ) )$. We assume that $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) > 0$ for all $( x , y ) \in \mathbb { R } ^ { 2 }$. Let $r ( x , y ) = \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) , s ( x , y ) = \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y )$ and $t ( x , y ) = \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y )$ so that, for all $( x , y ) \in \mathbb { R } ^ { 2 }$, $r ( x , y ) > 0$ and $r ( x , y ) t ( x , y ) - s ( x , y ) ^ { 2 } = 1$.

1. [IV.B.1)] Show that there exist two functions $\varphi$ and $\psi$ in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ such that $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \left\{ \begin{array} { l } \varphi ( u ( x , y ) , v ( x , y ) ) = x - \frac { \partial f } { \partial x } ( x , y ) \\ \psi ( u ( x , y ) , v ( x , y ) ) = - y + \frac { \partial f } { \partial y } ( x , y ) \end{array} \right.$$
2. [IV.B.2)] Compute $\frac { \partial \varphi } { \partial u } ( u ( x , y ) , v ( x , y ) ) , \frac { \partial \varphi } { \partial v } ( u ( x , y ) , v ( x , y ) ) , \frac { \partial \psi } { \partial u } ( u ( x , y ) , v ( x , y ) )$ and $\frac { \partial \psi } { \partial v } ( u ( x , y ) , v ( x , y ) )$ (which we will abbreviate as $\frac { \partial \varphi } { \partial u } , \frac { \partial \varphi } { \partial v } , \frac { \partial \psi } { \partial u }$ and $\frac { \partial \psi } { \partial v }$) in terms of $r ( x , y ) , s ( x , y )$ and $t ( x , y )$ (which we will abbreviate as $r , s$ and $t$).
3. [IV.B.3)] Show that $\frac { \partial \varphi } { \partial u }$ and $\frac { \partial \varphi } { \partial v }$ are bounded on $\mathbb { R } ^ { 2 }$.
4. [IV.B.4)] Show, using the first part, that $\frac { \partial \varphi } { \partial u }$ and $\frac { \partial \varphi } { \partial v }$ are constant.
5. [IV.B.5)] Deduce that $r , s$ and $t$ are constant.

Let $f$ be in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$: $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ We denote by $\mathcal { P } _ { 2 }$ the set of polynomial functions of degree $\leqslant 2$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$.
Show that the only functions in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$ belong to $\mathcal { P } _ { 2 }$.

Write a procedure or function in Maple or Mathematica that takes as input a quadruple $(a,b,c,d)$ of reals and returns, when possible, a real $t$ such that $\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) = R_t$ and an error message otherwise.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ and $(x,y,z)$ in $\mathbb{R}^3$, we denote by $\psi_A(x,y,z)$ the real part of the determinant of the matrix $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$, where $\mathrm{i}$ is the complex affix of the point $J = (0,1)$.
Specify the nature of the quadric $\mathcal{H}_A$ with equation $\psi_A(x,y,z) = 0$.

Show that $\Gamma^{s}(x_{0})$ is a vector subspace of $\mathcal{C}$, then that, for all real numbers $s_{1}$ and $s_{2}$ satisfying $0 \leq s_{1} \leq s_{2} < 1$, we have $\Gamma^{s_{2}}(x_{0}) \subset \Gamma^{s_{1}}(x_{0})$. Finally, determine $\Gamma^{0}(x_{0})$.
Recall: Let $x_{0} \in [0,1]$. For all $s \in [0,1[$, $\Gamma^{s}(x_{0})$ is the subset of $\mathcal{C}$ formed by functions $f$ which satisfy: $$\sup_{x \in [0,1] \backslash \{x_{0}\}} \frac{|f(x) - f(x_{0})|}{|x - x_{0}|^{s}} < +\infty .$$

Let $f \in \mathcal{C}$. If $f$ is differentiable at $x_{0}$, show that $f \in \Gamma^{s}(x_{0})$ for all $s \in [0,1[$.

Show that for all $x_{0} \in ]0,1[$, there exists $f \in \mathcal{C}$ non-differentiable at $x_{0}$ such that for all $s \in [0,1[$, $f \in \Gamma^{s}(x_{0})$.

Let $p : [0,1] \rightarrow \mathbf{R}$, $x \mapsto \sqrt{|1 - 4x^{2}|}$. Determine the pointwise Hölder exponent of $p$ at $\frac{1}{2}$.
Recall: For all $f \in \mathcal{C}$ and all $x_{0} \in [0,1]$, $$\alpha_{f}(x_{0}) = \sup \{s \in [0,1[ \mid f \in \Gamma^{s}(x_{0})\} .$$

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 $\omega_{f}$ 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 $$\omega_{f}(h') \leq \omega_{f}(h) + \omega_{f}(h' - h) .$$