Continuity at a Boundary

Q.

A real function f is said to be continuous if it is continuous at every point in ...... .

1. The domain of f

2. The range of f

3. Any interval of real numbers

4. [- ∞, ∞]

Q.

Which of the following is not continuous for all x ?

1. 
   

2. 
   

3. 
   

4. 
   

Q.

Discuss the continuity and differentiability of function f(x)=|x|+|x-1| in the interval (1, 2)

Q. Examine the continuity of the function $f\left(x\right)=\frac{x+6}{x^2-36},x\ne 6$

Q. Let $f\left(x\right)=\{\begin{array}{cc}\left[x\right]+[-x],& x\ne 2\\ \lambda ,& x=2\end{array};$ where $[.]$ denotes the greatest integer function. If $f$ is continuous at $x=2,$ then the value of $\lambda$ is

1. $-1$
2. $1$
3. $0$
4. $2$

Q. Range of the function $2y+3|x|=x$ is:

1. $R^{+}\cup \left\{0\right\}$
2. $R^{-}$
3. $R^{-}\cup \left\{0\right\}$
4. $R^{+}$

Q. If a function $f\left(x\right)$ is defined as $f\left(x\right)=\{\begin{array}{cc}-x,& x<0\\ x^2,& 0\le x\le 1\\ x^2-x+1,& x>1\end{array},$
then which among the following option is correct

1. $f\left(x\right)$ is differentiable function
2. $f\left(x\right)$ is not differentiable at $x=1$
3. $f\left(x\right)$ is not differentiable at $x=0$
4. $f\left(x\right)$ is not differentiable at $x=0,1$

Q.

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

1. Function will be continuous in any case

2. 

3. 

4. Function will be discontinuous in any case

Q. Let f(x) = $\{\begin{array}{c}1+x^3,x<0\\ x^2-1,x\ge 0\end{array}$ ; g(x) = $\{\begin{array}{c}(x-1{)}^{1/3},x<0\\ (x+1{)}^{1/2},x\ge 0\end{array}$. Discuss the continuity of g(f(x))

1. gof is discontinuous at x = 0
2. gof is discontinuous at x =-1
3. gof is discontinuous at x = 1
4. gof is continuous at x = 0

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

Q. $f\left(x\right)=$ $\frac{x^2-25}{x+5}$, $x$ $\ne$ $5$

Q. Discuss the continuity of the function $f$, where $f$ is defined by
$f\left(x\right)=$ $\{\begin{array}{c}3,if0\le x\le 1\\ 4,if1<x<3\\ 5,if3\le x\le 10\end{array}$

Q. If $f\left(x\right)=\{\begin{array}{cc}\frac{1}{5}(2x^2+3)& -\infty <x\le 1\\ 6-5x& 1<x<3\\ x-3& 3\le x<\infty \end{array}$

1. $f$ is continuous at $x=1$
2. $f$ is discontinuous at $x=1$
3. $f$ is continuous at $x=3$
4. $f$ is discontinuous at $x=2$

Q. If $f\left(x\right)=\{\begin{array}{ccc}\frac{1}{5}(2x^2+3)& -\infty <x\le 1& \\ 6-5x& 1<x<3& \\ x-3& 3\le x<\infty & \end{array}$, then $f$ is

1. discontinuous at $x=1$
2. discontinuous at $x=3$
3. continuous at $x=3$
4. continuous at $x=1$

Q. If $f\left(x\right)=\{\begin{array}{cc}{e}^x,& x<2\\ a+bx,& x\ge 2\end{array}$ is differentiable for all $x\in R,$ then which of the following is/are correct

1. $b=e^2$
2. $a+2b=e^2$
3. $a+b=0$
4. $a+b=e$

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. Let $a,b\in R,(a\ne 0).$ If the function $f$ defined as
$f\left(x\right)=\{\begin{array}{c}\frac{2x^2}{a},0\le x<1\\ \\ a,1\le x<\sqrt{2}\\ \\ \frac{2b^2-4b}{x^3},\sqrt{2}\le x<\infty \end{array}$
is continuous in the interval $[0,\infty ),$, then an ordered pair $(a,b)$ is:

1. $(-\sqrt{2},1-\sqrt{3})$
2. $(\sqrt{2},1-\sqrt{3})$
3. $(-\sqrt{2},1+\sqrt{3})$
4. $(\sqrt{2},-1+\sqrt{3})$

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, $f$ is :

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

Q. Which of the following functions is non-differentiable?

1. $f\left(x\right)=(e^x-1)|e^{2x}-1|$ defined in $R$
2. $f\left(x\right)=\frac{x-1}{x^2+1}$ defined in $R$
3. $f\left(x\right)\{\begin{array}{cc}||x-3|-1|,& x<3\\ \frac{x}{3}\left[x\right]-2,& x\ge 3\end{array}$ where [.] represents the greatest integer function
4. $f\left(x\right)=3(x-2{)}^{1/3}+3$ defined in $R$

Q. Discuss the continuity of f(x) in [0, 2] $f\left(x\right)=\{\begin{array}{cc}\left[\cos \pi x\right],& x\le 1\\ |2x-3|[x-2],& x>1\end{array}$ where $\left[t\right]$ represents the greatest integer function. Number of points at which it is discontinuous is

Q. Examine the following functions for continuity.

1. $f\left(x\right)=x-5$
2. $f\left(x\right)=\frac{1}{x-5}$
3. $f\left(x\right)=\frac{x^2-25}{x-5}$
4. $f\left(x\right)=|x-5|$

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. $Let\left(x\right)=\{\begin{array}{ccc}\frac{a|x^2-x-2|}{2+x-x^2},& x<2& \\ b,& x=2& \left(\right[.\left]denotesthegreatestintegerfunction\right)\\ \frac{x-\left[x\right]}{x-2},& x>2& \end{array}$
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. $A=(\sqrt{1-t^2}+t,0)$ and $B=(\sqrt{1-t^2}-t,2t)$ are two variable point where t is a parameter the locus of the middle point of AB is

1. a pair of straight line
2. a straight line
3. circle
4. none of these

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. Discuss the continuity to the given by
$f\left(x\right)=\{\begin{array}{c}x,\text{if}x\ge 0\\ x^2,\text{if}x<0\end{array}$

Q. $a=2,d=8,S_n=90,$ find n.

Q. Discuss the continuity of the function f, where f is defined by
f(x) = $\{\begin{array}{c}2x,ifx<0\\ 0,if0\le x\le 1\\ 4x,ifx>1\end{array}$

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. f(a) > f(b)

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

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

4. f(a) =