Continuity at a Boundary

Q.

The function $\frac{1}{1+x^2}$ is decreasing in the interval

1. $(-\infty ,1]$

2. $(-\infty ,0]$

3. $[1,\infty )$

4. $(0,\infty )$

Q.

The number of points at which the function $f\left(x\right)=\frac{1}{(x-[x\left]\right)}$ is not continuous is

1. $1$

2. $2$

3. $3$

4. none of these

Q.

Function f(x) is defined in [a, b]. If the function is continuous throughout the interval [a, b] then which among the following are correct

1. 

2. 

3. 

4. 

Q. Let $f:R\to R$ be a function defined as
$f\left(x\right)=\{\begin{array}{cc}5,& \text{if}x\le 1\\ a+bx,& \text{if}1<x<3\\ b+5x,& \text{if}3\le x<5\\ 30,& \text{if}x\ge 5\end{array}$
Then, which of the following is correct regarding the function $f$ :

1. continuous if $a=-5$ and $b=10$
2. continuous if $a=5$ and $b=5$
3. continuous if $a=0$ and $b=5$
4. not continuous for any values of $a$ and $b$

Q.

The function f(x) is continuous in the interval (a, b) then which among the following is true?

1. Function should exist at x=a but may not be continuous

2. Function may or may not have a value at x=a, b

3. f(a) > f(b)

4. f(a) =

Q.

If a function f(x) is defined in x $ϵ$ [a, b], then f(x) is continuous at a if

1. ${lim}_{x\to a}-f\left(x\right)={lim}_{x\to a^{+}}f\left(x\right)=f\left(a\right)$

2. ${lim}_{x\to a^{+}}f\left(x\right)=f\left(a\right)$

3. Function will be continuous in any case

4. Function will be discontinuous in any case

Q.

The function f(x) is continuous in the interval (a, b) then which among the following is true?

1. Function may or may not have a value at x=a, b

2. Function should exist at x=a but may not be continuous

3. f(a) = $\underset{x\to a^{+}}{lim}f\left(x\right)$

4. f(a) > f(b)

Q.

If a function f(x) is defined in x $ϵ$ [a, b], then f(x) is continuous at a if

1. 

2. 

3. Function will be continuous in any case

4. Function will be discontinuous in any case

Q. $Let\left(x\right)=\{\begin{array}{ccc}\frac{a|x^2-x-2|}{2+x-x^2},& x<2& \\ b,& x=2& \left(\right[.\left]denotesthegreatestintegerfunction\right)\\ \frac{x-\left[x\right]}{x-2},& x>2& \end{array}$
If f(x) is continuous at x = 2, then

1. a = 1, b = 2
2. a = 1, b = 1
3. a = 0, b = 1
4. a = 2, b = 1

Q. $Let\left(x\right)=\{\begin{array}{ccc}\frac{a|x^2-x-2|}{2+x-x^2},& x<2& \\ b,& x=2& \left(\right[.\left]denotesthegreatestintegerfunction\right)\\ \frac{x-\left[x\right]}{x-2},& x>2& \end{array}$
If f(x) is continuous at x = 2, then

1. a = 1, b = 2
2. a = 1, b = 1
3. a = 0, b = 1
4. a = 2, b = 1