Continuous Function

Q. $f\left(x\right)=[x{]}^3-[x^3]$, (where [.] is greatest integer function) is discontinuous at all

1. integers $n$
2. integers $n$ except $n=0\text{and}1$
3. integers $n$ except $n=0\text{and}1$, since $f\left(n^{-}\right)\ne f\left(n\right)$
4. integers $n$ except $n=0\text{and}1$, since $f\left(n^{+}\right)\ne f\left(n\right)$

Q. $f\left(x\right)=[x{]}^3-[x^3]$, (where [.] is greatest integer function) is discontinuous at all

1. integers $n$
2. integers $n$ except $n=0\text{and}1$
3. integers $n$ except $n=0\text{and}1$, since $f\left(n^{-}\right)\ne f\left(n\right)$
4. integers $n$ except $n=0\text{and}1$, since $f\left(n^{+}\right)\ne f\left(n\right)$

Q. $Letf:[-\frac{1}{3},3]\to Randg:[-\frac{1}{3},3]\to Rdefinedbyf\left(x\right)=[x^2-4]andg\left(x\right)=|x-2|f\left(x\right)+|3x-5|f\left(x\right)$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$ for $x\in R,$ then

1. f is discontinuous exactly at eight points in $[-\frac{1}{3},3]$
2. f is discontinuous exactly at nine points in $[-\frac{1}{3},3]$
3. g is discontinuous exactly at ten points in $[-\frac{1}{3},3]$
4. g is discontinuous exactly at nine points in $[-\frac{1}{3},3]$

Q. $f\left(x\right)$ is continuous function on [1, 3] and $f\left(1\right)=2,f\left(3\right)=-2,$ then which of the following not necessarily hold good?

1. $f\left(2\right)⩾0$
2. $x^{2}f\left(x\right)=0\text{has a root in}(1,3)$
3. $-2\le f\left(x\right)\le 2\forall x\in [1,3]$
4. $f\left(x\right)-x^2=0\text{has a root in}(1,3)$

Q. If $f\left(x\right)=\{\begin{array}{cc}x& \text{if}x\text{is rational}\\ 1-x& \text{if}x\text{is irrational}\end{array}$, then the number of points for $xϵR$ where $f\left(f\right(x\left)\right)$ is/are discontinuous

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