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}}
{x + \frac{1}{x},\;\;x \ne 0,} \\
{1,\;\;\;\;\;\;\;\,x = 0} \\
\end{array}} \right.
$
B: $f(x) = \left\{ {\begin{array}{*{20}{c}}
{(1 + {x^2})\frac{1}{{{x^2}}},\;\;x \ne 0} \\
{1,\;\;\;\;\;\;\;\;\;\quad \;\;x = 0} \\
\end{array}} \right.
$
C: $f(x) = [\cos x]
$($[\cdot]$表示取整函数)
D: $f(x) = {\mathop{\rm sgn}} (x)
$(符号函数)
A: $f(x) = \left\{ {\begin{array}{*{20}{c}}
{x + \frac{1}{x},\;\;x \ne 0,} \\
{1,\;\;\;\;\;\;\;\,x = 0} \\
\end{array}} \right.
$
B: $f(x) = \left\{ {\begin{array}{*{20}{c}}
{(1 + {x^2})\frac{1}{{{x^2}}},\;\;x \ne 0} \\
{1,\;\;\;\;\;\;\;\;\;\quad \;\;x = 0} \\
\end{array}} \right.
$
C: $f(x) = [\cos x]
$($[\cdot]$表示取整函数)
D: $f(x) = {\mathop{\rm sgn}} (x)
$(符号函数)
举一反三
- 8.下列函数在$x_0=0$处连续的为()。 A: $f(x) = \left\{ {\begin{array}{*{20}{c}}<br/>{{{\rm{e}}^{ - \frac{1}{{{x^2}}}}},\;\;x \ne 0} \\<br/>{0,\;\;\;\;\;x = 0} \\<br/>\end{array}} \right.<br/>$ B: $f(x) = [x]<br/>$ C: $f(x) = {\mathop{\rm sgn}} (\sin x)<br/>$ D: $f(x) = \left\{ {\begin{array}{*{20}{c}}<br/>{\frac{{\sin x}}{{\left| x \right|}},\;\;x \ne 0} \\<br/>{1,\;\;\;\;\;\;\;x = 0} \\<br/>\end{array}} \right.<br/>$
- 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: $f(x,y)=\left\{\begin{array}{cc}\dfrac{xy}{~x^2+y^2~},<br/>&x^2+y^2\neq 0; \\0, &x^2+y^2= 0. \end{array}\right.$
- 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: ${\rm{\{ }}\left. x \right|x \le 0\}<br/>$
- 8.下列函数中为无界函数的是 A: $f(x)=\frac{{{x}^{2}}+\sqrt{1+{{x}^{2}}}}{2+{{x}^{2}}},\ \quad x\in (-\infty ,+\infty )$ B: $f(x)=({\rm{sgn}}x)\cdot \sin \frac{1}{x},\quad x\ne 0$,${\rm{sgn}} x$为符号函数 C: $f(x)=\frac{[x]}{x},\quad x>0$,$[x]$为取整函数 D: $f(x)=\frac{x}{\ln x},\quad x\in (0,+\infty )$