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高二数学 | 高级数学
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12.5 Continuous Functions 连续函数


[1]

g(x) = \begin{cases} 2x-1 , 0 \leq x < 2 \\ 3 , x= 2 \\ x + 1 , 2 < x < 4 \\ x , x \geq 4 \end{cases},

For each of the number 2 and 4, check whether is continuous.

对于数字2和4,检查其是否连续。

Answer:
2连续,

4不连续

[2]

State the range of f (x) = \begin{cases} 3 - x  , x < -1 \\ 4  ,  x = -1  \\  2x + 2 ,  -1 < x < 1 \\ 3x + 1 , x \geq 1 \end{cases} where f is continuous
试写出值域其中f 为连续。

Answer:

R \ {-1}

[3]

g(x) = \begin{cases} 2x - x^2 , 0 \leq x \leq 2 \\ 2 - x , 2 < x < 3\\ x - 4 , 3 < x < 4\\ \pi , x \geq 4 \end{cases}

For each of the number 2, 3 and 4, verify whether is continuous.
State the range of g where it is continuous.

对每个数目 2,3和4, 检验其是否连续。
写出g连续的范围。

Answer:
2连续,3,4不连续;

(0, \infty) \ {4, 3}

[4]

Redefine the value of g(1) if g(x) = \frac{2x^2-x-1}{x-1} is continuous at x = 1.

重新定义 g(1) 的值若 g(x) = \frac{2x^2-x-1}{x-1} 在 x = 1 连续.

Answer:
3
[5]

Redefine the value of g(4) if g(x) = \frac{x+\sqrt x-6}{x-4} is continuous at x = 4.

重新定义 g(4) 的值若 g(x) = \frac{x+\sqrt x-6}{x-4} 在 x = 1连续.

Answer:
1\frac{1}{4}
[6]

Find the values of a and b that make h continuous on R, where

求 a 和 b 的值,使得 h 在 R连续, 其中

h(x) = \begin{cases} 2x + b , x < 2 \\ ax^2 - 1 , x > 2 \\ 2a+2b , x = 2  \end{cases}

Answer:
1\frac{1}{2} , 1