Let $f$ be the function defined by $f ( x ) = \sqrt { | x - 2 | }$ for all $x$. Which of the following statements is true?
(A) $f$ is continuous but not differentiable at $x = 2$.
(B) $f$ is differentiable at $x = 2$.
(C) $f$ is not continuous at $x = 2$.
(D) $\lim _ { x \rightarrow 2 } f ( x ) \neq 0$
(E) $x = 2$ is a vertical asymptote of the graph of $f$.

If the function $f$ is continuous at $x = 3$, which of the following must be true?
(A) $f ( 3 ) < \lim _ { x \rightarrow 3 } f ( x )$
(B) $\lim _ { x \rightarrow 3 ^ { - } } f ( x ) \neq \lim _ { x \rightarrow 3 ^ { + } } f ( x )$
(C) $f ( 3 ) = \lim _ { x \rightarrow 3 ^ { - } } f ( x ) = \lim _ { x \rightarrow 3 ^ { + } } f ( x )$
(D) The derivative of $f$ at $x = 3$ exists.
(E) The derivative of $f$ is positive for $x < 3$ and negative for $x > 3$.

For a function $y = f ( x )$ defined on all real numbers, let $N ( f )$ denote the smallest natural number $k$ such that the function $y = x ^ { k } f ( x )$ is continuous at $x = 0$. For example,
$$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { x } & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{array} \text { then } N ( f ) = 2 \right. \text { . }$$
For the following functions $g _ { i } ( i = 1,2,3 )$, let $N \left( g _ { i } \right) = a _ { i }$. Which correctly represents the order of $a _ { i }$? [3 points]
$$\begin{aligned} & g _ { 1 } ( x ) = \begin{cases} \frac { | x | } { x } & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{cases} \\ & g _ { 2 } ( x ) = \begin{cases} - x ^ { 2 } + 1 & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{cases} \\ & g _ { 3 } ( x ) = \begin{cases} \frac { 1 } { x ^ { 2 } } & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{cases} \end{aligned}$$
(1) $a _ { 1 } = a _ { 2 } < a _ { 3 }$
(2) $a _ { 1 } < a _ { 2 } = a _ { 3 }$
(3) $a _ { 1 } = a _ { 2 } = a _ { 3 }$
(4) $a _ { 2 } = a _ { 3 } < a _ { 1 }$
(5) $a _ { 3 } < a _ { 1 } = a _ { 2 }$

The function $$f(x) = \left\{ \begin{array}{cc} 5x + a & (x < -2) \\ x^{2} - a & (x \geq -2) \end{array} \right.$$ is continuous on the set of all real numbers. What is the value of the constant $a$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

We are given a function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ and we define a function $f_\xi : \mathcal{M}_d(\mathbb{R}) \rightarrow \mathcal{M}_d(\mathbb{R})$ such that $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad f_\xi(A) = \left(\xi\left(A_{i,j}\right)\right)_{1 \leqslant i,j \leqslant d}$$ We propose to determine the continuous functions $\xi : \mathbb{R} \rightarrow \mathbb{R}$ such that $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) \text{ invertible} \tag{V.1}$$
Determine the continuous functions $\xi$ satisfying condition (V.1) when $d = 1$.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Show that the function $\xi$ does not vanish on $\mathbb{R}^*$.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
The purpose of this question is to show $\xi(0) = 0$.
1) Show that if $\xi(0) \neq 0$, then there exists $\alpha > 0$ such that $\xi(0)\xi(2) = \xi(1)\xi(\alpha)$.
2) Conclude.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Let $\eta = \xi^{-1} : I \rightarrow \mathbb{R}$ be the inverse function of the bijection $\xi : \mathbb{R} \rightarrow I$. Show that where it is defined $$(\eta(xy))^2 = \eta\left(x^2\right)\eta\left(y^2\right)$$

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), and $\eta = \xi^{-1} : I \rightarrow \mathbb{R}$ is the inverse function of the bijection $\xi : \mathbb{R} \rightarrow I$.
We assume in this question that the function $\eta$ takes strictly positive values on $I \cap {]0, +\infty[}$.
1) Show that the function $f = \ln \circ \eta \circ \exp$ satisfies equation (IV.1) on an interval $]-\infty, M[$, with $M$ (possibly infinite) to be determined as a function of the interval $I$.
2) Deduce that on the interval $I \cap {]0, +\infty[}$ the function $\eta$ is of the form $$\eta : x \mapsto K_1 x^{\alpha_1}$$ with two constants $K_1 > 0$ and $\alpha_1 > 0$.
3) Show that on the interval $I \cap {]-\infty, 0[}$ the function $\eta$ is of the form $$\eta : x \mapsto K_2(-x)^{\alpha_2}$$ with two constants $K_2 < 0$ and $\alpha_2 > 0$.
4) Show that $I = \mathbb{R}$ then that the function $\eta$ is an odd function.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Deduce in the general case that, if $\xi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function satisfying condition (V.1), then it is odd and its restriction to $\mathbb{R}_+^*$ is of the form $x \mapsto Cx^\beta$, with $C \neq 0$ and $\beta > 0$.

Justify that the mapping $f$ establishes a bijection from the interval $\left[ - 1 , + \infty \left[ \right. \right.$ onto the interval $\left[ - e ^ { - 1 } , + \infty [ \right.$, where $f : \mathbb{R} \rightarrow \mathbb{R}$, $x \mapsto x\mathrm{e}^{x}$.

Let $f(x) = xe^x$. Prove that the mapping $f$ establishes a bijection from the interval $] - \infty , - 1 ]$ onto the interval $\left[ - \mathrm { e } ^ { - 1 } , 0 [ \right.$. In the rest of the problem, the inverse of this bijection is denoted $V$.

Let $I$ be an open interval of $\mathbb { R }$. We are given a function $f : I \rightarrow \mathbb { R }$ of class $\mathcal { C } ^ { 3 }$, such that $f ^ { \prime } ( x ) > 0$ for all $x \in I$. Show that $f$ is bijective from $I$ onto the open interval $f ( I )$.
We denote by $g : f ( I ) \rightarrow I$ its inverse function. Recall the value of $g ^ { \prime } ( f ( x ) )$. Express $g ^ { \prime \prime } ( f ( x ) )$ as a function of the successive derivatives of $f$ at $x$.

Prove that the endomorphism $D - I$ is invertible and express $L$ in terms of $(D-I)^{-1}$, where $L$ is defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$.

Consider the following two statements: (I) There exists a differentiable function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $g\left(x^3 + x^5\right) = e^x - 100$. (II) There exists a continuous function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $g\left(e^x\right) = x^3 + x^5$. Then
(A) Only (I) is correct.
(B) Only (II) is correct.
(C) Both (I) and (II) are correct.
(D) Neither (I) nor (II) is correct.

Let $f: \{1,2,3,4\} \to \{1,2,3,4\}$ and $g: \{1,2,3,4\} \to \{1,2,3,4\}$ be invertible functions such that $f \circ g = $ identity. Then
(A) $f = g^{-1}$
(B) $g = f^{-1}$
(C) $f \circ g \neq g \circ f$
(D) $f \circ g = g \circ f$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions satisfying $$f(x + y) = f(x) + f(y) + f(x)f(y) \text{ and } f(x) = xg(x)$$ for all $x, y \in \mathbb{R}$. If $\lim_{x \rightarrow 0} g(x) = 1$, then which of the following statements is/are TRUE?
(A) $f$ is differentiable at every $x \in \mathbb{R}$
(B) If $g(0) = 1$, then $g$ is differentiable at every $x \in \mathbb{R}$
(C) The derivative $f'(1)$ is equal to 1
(D) The derivative $f'(0)$ is equal to 1

Let $S = ( 0,1 ) \cup ( 1,2 ) \cup ( 3,4 )$ and $T = \{ 0,1,2,3 \}$. Then which of the following statements is(are) true?
(A) There are infinitely many functions from $S$ to $T$
(B) There are infinitely many strictly increasing functions from $S$ to $T$
(C) The number of continuous functions from $S$ to $T$ is at most 120
(D) Every continuous function from $S$ to $T$ is differentiable