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Data publicarii: 13 Martie, 2009

TEORIE

Functia modul (valoare absoluta):

f:R - > [0,+oo),

$f(x)=|x|=\begin{cases}-x,x\in{(-\infty,0)}\\x,x\in{[0,+\infty)}\end{cases}$ f(x)=|x|=\begin{cases}-x,x\in{(-\infty,0)}\\x,x\in{[0,+\infty)}\end{cases}

$f(x)=|x|=\begin{cases}-x,x\in{(-\infty,0]}\\x,x\in{(0,+\infty)}\end{cases}$ f(x)=|x|=\begin{cases}-x,x\in{(-\infty,0]}\\x,x\in{(0,+\infty)}\end{cases}

$f(x)=|x|=\begin{cases}-x,x\in{(-\infty,0)}\\{0},{x=0}\\{x},{x}\in{(0,+\infty)}\end{cases}.$ f(x)=|x|=\begin{cases}-x,x\in{(-\infty,0)}\\{0},{x=0}\\{x},{x}\in{(0,+\infty)}\end{cases}.

Proprietati:

|x| > 0 sau |x| = 0, oricare ar fi x real;
|x| = 0 <=> x = 0;
|x|² = 0, oricare ar fi x real;
|x·y| = |x|·|y|, oricare ar fi x si y reali => |- x| = |x|, oricare ar fi x real;
|x/y| = |x|/|y|, oricare ar fi x si y reali, y nenul;
${|x|-|y|}\le{|{x}\pm{y}|}\le{|x|+|y|},\;\forall{x,y}\in{\mathbb{R}};$ {|x|-|y|}\le{|{x}\pm{y}|}\le{|x|+|y|},\;\forall{x,y}\in{\mathbb{R}};
|x| = a <=> x = a sau x = - a, unde a > 0;
|x| = |y| <=> x = y sau x = - y;
|x| < c < = > x € (- c, c), oricare ar fi c > 0;
|x| > c < = > x € (- 00,- c)U(c,+ 00), oricare ar fi c > 0.

Functia parte intreaga:

f:R - > Z, f(x) = [x],

unde prin [x] înţelegem cel mai mare număr întreg n, care este mai mic sau egal cu x,

adică [x] = n, dacă

$n\leq{x}<{n+1},n\in{\mathbb{Z}}.$ n\leq{x}<{n+1},n\in{\mathbb{Z}}.

Observatii:

1) Din definitie rezulta imediat dubla inegalitate remarcabila:

$[x]\leq{x}<{[x]+1},\forall{x}\in{\mathbb{R}}.$ [x]\leq{x}<{[x]+1},\forall{x}\in{\mathbb{R}}. $\Rightarrow$ \Rightarrow ${x-1}<{[x]}\leq{x},\forall{x}\in{\mathbb{R}}.$ {x-1}<{[x]}\leq{x},\forall{x}\in{\mathbb{R}}.

2) Functia se expliciteaza astfel:

f(x) = [x] = $\begin{cases}\cdots\\-k,x\in{[-k,-k+1)}\\\cdots\\-3,x\in{[-3,-2)}\\-2,x\in{[-2,-1)}\\-1,x\in{[-1,0)}\\{0},x\in{[0,1)}\\1,x\in{[1,2)}\\2,x\in{[2,3)}\\3,x\in{[3,4)}\\\cdots\\k,x\in{[k,k+1)}\\\cdots\end{cases},k\in{\mathbb{N}}.$ \begin{cases}\cdots\\-k,x\in{[-k,-k+1)}\\\cdots\\-3,x\in{[-3,-2)}\\-2,x\in{[-2,-1)}\\-1,x\in{[-1,0)}\\{0},x\in{[0,1)}\\1,x\in{[1,2)}\\2,x\in{[2,3)}\\3,x\in{[3,4)}\\\cdots\\k,x\in{[k,k+1)}\\\cdots\end{cases},k\in{\mathbb{N}}.

3) [x] + [x + (1/n)] + [x + (2/n)] + ... + [x + (n - 1)/n] = [nx]; n € N*,

identitatea lui Hermite.

Proprietati:

Pentru orice numere reale x si y si orice k intreg avem:

x = [x] < = > x € Z;
(x < y sau x = y) = > ([x] < [y] sau [x] = [y]);
([x + y] > [x] + [y]) sau ([x + y] = [x] + [y]);
[x + k] = [x] + k.

Functia parte fractionara (zecimala):

f:R - > [0;1), f(x) = {x}, unde, prin definitie, {x} = x - [x].

Explicitarea acestei functii este:

$f(x)=\begin{Bmatrix}x\end{Bmatrix}=\begin{cases}\cdots\\x+k,x\in[-k,-k+1)\\\cdots\\x+2,x\in{[-2,-1)}\\x+1,x\in{[-1,0)}\\x,x\in{[0,1)}\\x-1,x\in{[1,2)}\\x-2,x\in{[2,3)}\\\cdots\\x-k,x\in{[k,k+1)}\\\cdots\end{cases},k\in{\mathbb{N}}.$ f(x)=\begin{Bmatrix}x\end{Bmatrix}=\begin{cases}\cdots\\x+k,x\in[-k,-k+1)\\\cdots\\x+2,x\in{[-2,-1)}\\x+1,x\in{[-1,0)}\\x,x\in{[0,1)}\\x-1,x\in{[1,2)}\\x-2,x\in{[2,3)}\\\cdots\\x-k,x\in{[k,k+1)}\\\cdots\end{cases},k\in{\mathbb{N}}.

Proprietati:

{x} = 0 < = > x € Z;
{x + k} = {x}.

Functia semn (signum, signatura):

f:R - > {- 1, 0, 1},

$f(x)=sgn(x)=\begin{cases}-1,x\in{(-\infty,0)}\\{0},x=0\\1,x\in{(0,+\infty)}\end{cases}.$ f(x)=sgn(x)=\begin{cases}-1,x\in{(-\infty,0)}\\{0},x=0\\1,x\in{(0,+\infty)}\end{cases}.

Observatie:

Din definitie rezulta ca pentru orice x real nenul: sgn(x) = x / |x|.

Functia lui Dirichlet:

f:R - > {0; 1},

$f(x)=\begin{cases}1,x\in{\mathbb{Q}}\\{0},x\in{\mathbb{R}\setminus{\mathbb{Q}}}\end{cases}.$ f(x)=\begin{cases}1,x\in{\mathbb{Q}}\\{0},x\in{\mathbb{R}\setminus{\mathbb{Q}}}\end{cases}.

Functia caracteristica a unei multimi:

Fie o multime M de numere reale; numim functia caracteristica a multimii M

(notata de obicei fM, functia fM:R - > {0;1},

$f_M(x)=\begin{cases}1,x\in{M}\\{0},x\in{\mathbb{R}}\setminus{M}\end{cases}.$ f_M(x)=\begin{cases}1,x\in{M}\\{0},x\in{\mathbb{R}}\setminus{M}\end{cases}.