绝对值函数
$y=\left|x\right|=
\left\{\begin{matrix}
x, x \ge 0 &\\
-x, x < 0 &
\end{matrix}\right.$
nature:
$\left|x\right|=x \Leftrightarrow x \ge 0，\left|x\right|=-x \Leftrightarrow x \le 0$
Graphics:
Rounding function
$ y = [x] = $ largest integer less than or equal to $ X $ is
represented by piecewise function: $ y = [x] = n, n \ le x <n + 1 $ ($ n $ is an integer )
nature:
$ [X] \ le x <[x] + 1, [x] = x \ Leftrightarrow x $ is an integer, $ [x + y] \ ge [x] + [y], [x + n] = [x ] + n $ ($ n $ is an integer)
Graphics :( step curve)
符号函数
$y=sgnx=
\left\{\begin{matrix}
1,& x > 0 \\
0,& x = 0 \\
-1,& x < 0
\end{matrix}\right.$
性质：
$sgnx=1 \Leftrightarrow x > 0, sgnx=-1 \Leftrightarrow x < 0$
$sgn(x-a) = 1 \Leftrightarrow x > a, sgn(x-a) = -1 \Leftrightarrow x < a$
$x=sgnx \cdot \left|x\right|,\left|x\right|=sgnx \cdot x$
Graphics:
Dirichlet function
$ Y = D (X) =
\ left \ {\ the begin Matrix {}
. 1, is a rational number X & \\
0, X is an irrational number &
\ end {matrix} \ right. $
Properties:
Dirichlet function has many bad properties
1) no graphics Dirichlet function (no curve segment)
2) n Dirichlet function in any period of the periodic function is a rational number, it is no minimum positive period
3 ) Dirichlet function everywhere without limit, not everywhere continuous, always non-conductive, non-integrable on any interval
Dirichlet function used to give trans configurations and embodiments having a particular quality function
as a function of: $ y = xD (x) $ only continuous, discontinuous at other points at the origin,
the function $ y = x ^ {2} D (x) $ may be turned only at the origin, intermittently (and thus non-conductive) at the other point
note:
Dirichlet function may be defined as the limit of $ D (x) = \ lim_ {m \ rightarrow \ infty} [\ lim_ {n \ rightarrow \ infty} cos ^ {n} (\ pi m! X)] $