Let $f : \mathbb{R} \longrightarrow \mathbb{R}$ be defined as $$f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x^2}\right), & \text{if } x \neq 0 \\ 0, & \text{otherwise} \end{cases}$$ Choose the correct statement(s) from below:
(A) $f$ is continuous;
(B) $f$ is discontinuous at 0;
(C) $f$ is differentiable;
(D) $f$ is continuously differentiable.

For a quartic function $f ( x )$ with leading coefficient 1, the function $g ( x )$ satisfies the following conditions. (가) When $- 1 \leqq x < 1$, $g ( x ) = f ( x )$. (나) For all real numbers $x$, $g ( x + 2 ) = g ( x )$.
Which of the following statements in are correct? [4 points]
Remarks ㄱ. If $f ( - 1 ) = f ( 1 )$ and $f ^ { \prime } ( - 1 ) = f ^ { \prime } ( 1 )$, then $g ( x )$ is differentiable on the entire set of real numbers. ㄴ. If $g ( x )$ is differentiable on the entire set of real numbers, then $f ^ { \prime } ( 0 ) f ^ { \prime } ( 1 ) < 0$. ㄷ. If $g ( x )$ is differentiable on the entire set of real numbers and $f ^ { \prime } ( 1 ) > 0$, then there exists $c$ in the interval $( - \infty , - 1 )$ such that $f ^ { \prime } ( c ) = 0$.
(1) ᄀ
(2) ᄂ
(3) ᄀ, ᄃ
(4) ㄴ,ㄷ
(5) ᄀ, ᄂ, ᄃ

For the function $f ( x ) = e ^ { x + 1 } - 1$ and a natural number $n$, let the function $g ( x )$ be defined as $$g ( x ) = 100 | f ( x ) | - \sum _ { k = 1 } ^ { n } \left| f \left( x ^ { k } \right) \right|$$ Find the sum of all natural numbers $n$ such that $g ( x )$ is differentiable on the entire set of real numbers. [4 points]

Let $n \in \mathbb{N}$.
a) Show that the function $F_n$ is of class $C^\infty$ on $\mathbb{R}$.
b) For $x \in ]-1,1[$, give a simple expression for $F_n'(x)$. Justify the calculation carefully.

Let $M \in \mathbb{R}_+^* \cup \{+\infty\}$ and $f : {]-\infty, M[} \rightarrow \mathbb{R}$ be a continuous function such that $$\forall (x, y) \in {\left]-\infty, \frac{M}{2}\right[}^2, \quad 2f(x+y) = f(2x) + f(2y) \tag{IV.1}$$
Deduce that the function $f$ is of class $C^\infty$ on $]-\infty, M[$.

Show that the derivative of every element of $\mathcal{D}$ is an element of $\mathcal{D}$.

For every function $f$ belonging to $\mathcal{F}_{sr}$ and every non-zero natural number $n$, we set $$\left(f * \rho_n\right)(x) = \int_{\mathbb{R}} f(t) \rho_n(x-t) \mathrm{d}t$$
Let $f$ be a function belonging to $\mathcal{F}_{sr}$. Show that the function $f * \rho_n$ is of class $\mathcal{C}^{\infty}$.

We say that a real function $f$ of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$ has rapid decay if $$\forall (n,m) \in \mathbb{N}^2, \lim_{x \rightarrow +\infty} x^m f^{(n)}(x) = \lim_{x \rightarrow -\infty} x^m f^{(n)}(x) = 0$$ We denote $\mathcal{S}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ with rapid decay.
Show that if $f$ is in $\mathcal{S}$ then $f^{(p)}$ is in $\mathcal{S}$ for every natural number $p$.

Verify that $\varphi$ is of class $\mathscr{C}^{0}$ on $[0, +\infty[$ and $\mathscr{C}^{\infty}$ on $]0, +\infty[$. Give the limit of the derivative $\varphi'(t)$ of $\varphi$ as $t$ tends to 0 in $]0, +\infty[$.
Where $\varphi$ is defined by $$\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$$

For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by
$$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$
and the sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ defined by
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
IV.A.1) Show that the function $g$ is of class $C^{1}$ on $]-1,1[\backslash\{0\}$ and continuous on $]-1,1[$.
IV.A.2) Calculate the limit of $g'$ at 0. Deduce that $g$ is of class $C^{1}$ on $]-1,1[$.

Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. Let $h$ be the function defined on $\mathbb{R}$, which is 1-periodic and which equals $\mathcal{F}(f)$ on the interval $[-1/2, 1/2]$. Show that $h$ is of class $C^{\infty}$ on $\mathbb{R}$.

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Calculate $\lim _ { \substack { x \rightarrow 1 \\ x < 1 } } \varphi ^ { \prime } ( x )$ and demonstrate that $\varphi$ is of class $C ^ { 1 }$ on $\mathbb { R }$.

We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Demonstrate that $\theta$ is of class $C ^ { \infty }$ on $\mathbb { R }$.

Let $\varphi$ be the function defined on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \quad \varphi(t) = \begin{cases} 0 & \text{if } t \leqslant 0 \\ \mathrm{e}^{-1/t} & \text{otherwise} \end{cases}$$
Show that $\varphi$ is of class $\mathcal{C}^\infty$ on $\mathbb{R}$.
One may show that: $\forall k \in \mathbb{N}, \exists P_k \in \mathbb{R}[X], \forall t > 0, \varphi^{(k)}(t) = P_k(1/t)\mathrm{e}^{-1/t}$.

Let $\psi$ be the function defined on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \quad \psi(t) = \begin{cases} 0 & \text{if } t \notin ]-1,1[ \\ \mathrm{e}^{1/(t^2-1)} & \text{otherwise.} \end{cases}$$
Show, by expressing it in terms of $\varphi$, that $\psi$ is of class $\mathcal{C}^\infty$.

Let $\theta$ be the unique antiderivative of $\psi$ vanishing at 0. Show that $\theta$ is of class $\mathcal{C}^\infty$, constant on $]-\infty, -1]$ (we denote this constant by $A$) and constant on $[1, +\infty[$ (we denote this constant by $B$). Verify that $A \neq B$.

Let $\varphi$ be the function defined on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \quad \varphi(t) = \begin{cases} 0 & \text{if } t \leqslant 0 \\ \mathrm{e}^{-1/t} & \text{otherwise} \end{cases}$$ Show that $\varphi$ is of class $\mathcal{C}^\infty$ on $\mathbb{R}$.
One may show that: $\forall k \in \mathbb{N}, \exists P_k \in \mathbb{R}[X], \forall t > 0, \varphi^{(k)}(t) = P_k(1/t)\mathrm{e}^{-1/t}$.

Let $\psi$ be the function defined on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \quad \psi(t) = \begin{cases} 0 & \text{if } t \notin \left]-1,1\right[ \\ \mathrm{e}^{1/(t^2-1)} & \text{otherwise.} \end{cases}$$ Show, by expressing it in terms of $\varphi$, that $\psi$ is of class $\mathcal{C}^\infty$.

Let $\theta$ be the unique antiderivative of $\psi$ vanishing at 0. Show that $\theta$ is of class $\mathcal{C}^\infty$, constant on $]-\infty, -1]$ (we denote this constant by $A$) and constant on $[1, +\infty[$ (we denote this constant by $B$). Verify that $A \neq B$.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by $$f_A(t) = \operatorname{det}(A + tM).$$ Show that $f_A$ is of class $C^\infty$ on $\mathbf{R}$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
Show that $f _ { A }$ is of class $C ^ { \infty }$ on $\mathbf { R }$.

Let $f(x) = e^{-1/x}$ for $x > 0$ and $f(x) = 0$ for $x \leq 0$. Which of the following is true?
(A) $f$ is not differentiable at $x = 0$
(B) $f$ is differentiable at $x = 0$ but $f'$ is not differentiable at $x = 0$
(C) $f$ is differentiable at $x = 0$ and $f'$ is differentiable at $x = 0$
(D) $f$ is differentiable everywhere and $f'$ is also differentiable everywhere

Let $f(x) = x|x|^n$ for $n \geq 1$ a positive integer. Which of the following is true?
(A) $f$ is differentiable everywhere except at $x = 0$
(B) $f$ is continuous but not differentiable at $x = 0$
(C) $f$ is differentiable everywhere
(D) None of the above