8.下列函数在$x_0=0$处连续的为()。
A: $f(x) = \left\{ {\begin{array}{*{20}{c}}
{{{\rm{e}}^{ - \frac{1}{{{x^2}}}}},\;\;x \ne 0} \\
{0,\;\;\;\;\;x = 0} \\
\end{array}} \right.
$
B: $f(x) = [x]
$
C: $f(x) = {\mathop{\rm sgn}} (\sin x)
$
D: $f(x) = \left\{ {\begin{array}{*{20}{c}}
{\frac{{\sin x}}{{\left| x \right|}},\;\;x \ne 0} \\
{1,\;\;\;\;\;\;\;x = 0} \\
\end{array}} \right.
$
A: $f(x) = \left\{ {\begin{array}{*{20}{c}}
{{{\rm{e}}^{ - \frac{1}{{{x^2}}}}},\;\;x \ne 0} \\
{0,\;\;\;\;\;x = 0} \\
\end{array}} \right.
$
B: $f(x) = [x]
$
C: $f(x) = {\mathop{\rm sgn}} (\sin x)
$
D: $f(x) = \left\{ {\begin{array}{*{20}{c}}
{\frac{{\sin x}}{{\left| x \right|}},\;\;x \ne 0} \\
{1,\;\;\;\;\;\;\;x = 0} \\
\end{array}} \right.
$
举一反三
- 6.下列函数中$x=0$是其可去间断点的为()。 A: $f(x) = \left\{ {\begin{array}{*{20}{c}}<br/>{x + \frac{1}{x},\;\;x \ne 0,} \\<br/>{1,\;\;\;\;\;\;\;\,x = 0} \\<br/>\end{array}} \right.<br/>$ B: $f(x) = \left\{ {\begin{array}{*{20}{c}}<br/>{(1 + {x^2})\frac{1}{{{x^2}}},\;\;x \ne 0} \\<br/>{1,\;\;\;\;\;\;\;\;\;\quad \;\;x = 0} \\<br/>\end{array}} \right.<br/>$ C: $f(x) = [\cos x]<br/>$($[\cdot]$表示取整函数) D: $f(x) = {\mathop{\rm sgn}} (x)<br/>$(符号函数)
- 5.下列函数中,在其定义域上有最大值和最小值的是()。 A: $f(x)=\left\{ \begin{array}{*{35}{l}} \ln \left| x \right|,\ \ \ x\ne 0 \\ 0,\ \ \ \ \ \ \ \ x=0 \\ \end{array} \right.$ B: $f(x)=\ln \left( \left| x \right|+1 \right)\ x\in [-1,1]$ C: $f(x)=\ln \left| x \right|,\ \ \ x\in [-1,1]\backslash \{0\}$ D: $f(x)=\left\{ \begin{array}{*{35}{l}} \ln \left| x \right|,\ \ \ 0\lt |x|\lt 1 \\ 0,\ \ \ \ \ \ \ \ x=0 \\ \end{array} \right.$
- 下列函数是多元初等函数的是( ) A: $f(x,y)=\left|x+y\right|$; B: $f(x,y)=\text{sgn}(x+y)$; C: $f(x,y)=\dfrac{\arcsin<br/>x-e^{y}}{~\ln(x^2+y^2)~}$; D: $f(x,y)=\left\{\begin{array}{cc}\dfrac{xy}{~x^2+y^2~},<br/>&x^2+y^2\neq 0; \\0, &x^2+y^2= 0. \end{array}\right.$
- 设随机变量$X$的概率密度为$f(x)=\left\{\begin{array}{left}e^{-x},& x\ge 0\\0 ,&x<0\end{array}\right.$,则$E(e^{-2X})=$
- 函数$y = \arcsin (2x + 1)<br/>$的定义域为 ( ). A: $\{ \left. x \right| - 1 \le x \le 0\} <br/>$ B: $\{ \left. x \right| - \frac{1}{2} \le x \le 0\} <br/>$ C: $\{ \left. x \right|x \ge - \frac{1}{2}\} <br/>$ D: ${\rm{\{ }}\left. x \right|x \le 0\}<br/>$