Points of Discontinuity

Q.

Define left-hand limit.

Q. Let $\left[t\right]$ denotes the greatest integer $\le t$ and $\underset{x\to 0}{lim}x\left[\frac{4}{x}\right]=A.$ Then the function, $f\left(x\right)=\left[x^2\right]\sin \pi x$ is discontinuous, when $x$ is equal to

1. $\sqrt{A+1}$
2. $\sqrt{A}$
3. $\sqrt{A+5}$
4. $\sqrt{A+21}$

Q. Let $\left[t\right]$ denote the greatest integer $\le t.$ The number of points where the function $f\left(x\right)=\left[x\right]|x^2-1|+\sin \left(\frac{\pi }{\left[x\right]+3}\right)-[x+1],x\in (-2,2)$ is not continuous is

Q.

The function $f:\mathrm{ℂ}\to \mathrm{ℂ}$ defined by $f\left(x\right)=\frac{ax+b}{cx+d}$ for $x\in \mathrm{ℂ}$, where $bd\ne 0$, reduces to a constant function if

1. $a=c$

2. $b=d$

3. $ad=bc$

4. $ab=cd$

Q. Let $\left[y\right]$ and $\left\{y\right\}$ denote the greatest integer less than or equal to $y$ and fractional part of $y$ respectively. Then the number of points of discontinuity of the function $f\left(x\right)=\left[5x\right]+\left\{3x\right\}$ in $[0,5]$ is

Q. Number of point(s) where the function $f\left(x\right)=max(\sqrt{2x-x^2},2-x)$ is non-differentiable, is/are

1. 0
2. 1
3. 2
4. 3

Q.

What are the properties of limits?

Q. The point(s) of discontinuity of the composite function y = f (f(x)) , where $f\left(x\right)=\frac{1}{x-1}$ is/are

1. x = 1
2. x = 0
3. x = 2
4. x = -2

Q. Let $f:[-1,3]\to R$ be defined as

$f\left(x\right)=\{\begin{array}{c}|x|+\left[x\right],-1\le x<1\\ x+|x|,1\le x<2\\ x+\left[x\right],2\le x\le 3\end{array}$

where $\left[t\right]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is discontinuous at :

1. only one point
2. only two points
3. only three points
4. four or more points

Q.

What is a strictly decreasing function?

Q. Find all point of discontinuity of the function $f\left(t\right)=\frac{1}{t^2+t-2},\mathrm{where}t=\frac{1}{x-1}$

Q. The number of points of discontinuity of $f\left(x\right)$ where $f\left(x\right)=|||x+\left[x\right]|-3\left[x\right]|-5\left[x\right]|\text{on}[-2,2]$ is
(where $\left[x\right]$ denotes greatest integer function)

1. 2
2. 4
3. 5
4. 6

Q.

Find point local maxima for the function f(x)=x³ +x² +x+1

1. x=0

2. No point of local maxima

3. x=1

4. x=3

Q. Which of the following functions is concave down over the set of positive real numbers:

1. $f\left(x\right)=2^x$
2. $f\left(x\right)=x^{\frac{1}{2}}$
3. $f\left(x\right)=3x+2$
4. $f\left(x\right)=x^3$

Q. Let $f:R\to R$ be a function defined by $f\left(x\right)=max\{x,x^2\}$. Let $S$ denote the set of all points in $R$, where $f$ is not differentiable. Then

1. $\{0,1\}$
   
2. $\varphi$ (an empty set)
   
3. $\left\{1\right\}$
   
4. $\left\{0\right\}$
   

Q. The number of point(s) of discontinuity of $f\left(x\right)=\left[5x\right],x\in [0,1]$ is

Q. The number of points of discontinuity of $f\left(x\right)$ where $f\left(x\right)=|||x+\left[x\right]|-3\left[x\right]|-5\left[x\right]|\text{on}[-2,2]$ is
(where $\left[x\right]$ denotes greatest integer function)

1. 2
2. 4
3. 5
4. 6

Q. The set of all points, where the function $f\left(x\right)=\sqrt{1-e^{-x^2}}$ is differentiable, is

1. $(-1,\infty )$
2. $(0,\infty )$
3. $(-\infty ,\infty )$
4. $(-\infty ,0)\cup (0,\infty )$

Q. The points where the function $f\left(x\right)=\left[x\right]+|1-x|,-1\le x\le 3$, where $[.]$ denotes the greatest integer function, is not differentiable are

1. $x=-1,0,1,2,3$
2. $x=0,1,2,3$
3. $x=-1,0,1,2$
4. $x=-1,0,2$

Q. If $y=f\left(x\right)$ is a differentiable function, then show that $\frac{d^{2}x}{d{y}^2}=-{\left(\frac{dy}{dx}\right)}^{-3}\frac{d^{2}y}{d{x}^2}$

Q. Find the square root of the given below complex number:
(ii) $-7-24i$

Q. $y=\frac{1}{t^2+t-2}$ where $t=\frac{1}{x-1}$ then the number of points of discontinuities of $y=f\left(x\right),xϵR$ is

1. 1
2. 2
3. 3
4. Infinite

Q.

n(n+1)(n+5) is a multiple of 3

Please give the answers of this question. I hAve seen the solutions but I am not able to understand the last k+1 part.

Q. Let $f:[-1,3]\to R$ be defined as

$f\left(x\right)=\{\begin{array}{c}|x|+\left[x\right],-1\le x<1\\ x+|x|,1\le x<2\\ x+\left[x\right],2\le x\le 3\end{array}$

where $\left[t\right]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is discontinuous at :

1. only one point
2. only two points
3. only three points
4. four or more points

Q. Find all the points of discontinuity of $f$ defined by $f\left(x\right)=\left|x\right|-|x+1|.$

Q. Let $\left[t\right]$ denotes the greatest integer less than or equal to $t$ and $\underset{x\to 0}{lim}x\left[\frac{4}{x}\right]=A.$ If function $f\left(x\right)=\left[x^2\right]\sin \pi x$ is discontinuous, then possible value of $x$ is

1. $\sqrt{A+1}$
2. $\sqrt{A}$
3. $\sqrt{A+5}$
4. $\sqrt{A+21}$

Q. Let $f\left(x\right)=\cos x$ and $g\left(x\right)=[x+2]$, where $[.]$ denotes the greatest integer function. Then, ${\left(gof\right)}^{\prime }\left(\frac{\pi }{2}\right)$ is?

1. $0$
2. $1$
3. $-1$
4. Does not exist

Q. Which of the following functions is concave down over the set of positive real numbers:

1. $f\left(x\right)=2^x$
2. $f\left(x\right)=x^{\frac{1}{2}}$
3. $f\left(x\right)=3x+2$
4. $f\left(x\right)=x^3$

Q. If $f\left(x\right)=\frac{1}{1-x}$, then the set of points discontinuity of the function f (f(f(x))) is
(a) {1}
(b) {0, 1}
(c) {−1, 1}
(d) none of these

Q. Let $f\left(x\right)=\{\begin{array}{cc}\underset{n\to \infty }{lim}\left(\frac{|x+1{|}^n+x^2}{|x|+x^{2n}}\right);& -6\le x<0\\ \left\{\sin x\right\};& 0\le x\le 6\end{array}$ where $\left\{k\right\}$ denotes the fractional part of $k.$
Then number of points at which $f$ is not differentiable in $(-6,6)$ is equal to