Question

Let $f:\left[0, \frac{5}{2}\right) \rightarrow\left(\frac{1}{2}, 3\right]$ be a function defined as $f(x)=\frac{[x]+1}{\{x\}+1}$ where $[ \,\,\,]$ and $\{\,\,\,\}$ represent the greatest integer function and fractional part. Given that $\min \cdot\left(\displaystyle\lim _{x \rightarrow 1^{-}} f(x), \displaystyle\lim _{x \rightarrow 1^{+}} f(x)\right)=\frac{m}{p}$, G.C.D. $(m, p)=1$ and maximum of the values of $x$, at which $f$ is discontinuous, is equal to $f(k) .$ Find the value of $p^{2}-k^{2} m^{2} .$

Answer 1

$f$ is discontinuous at $x=1,2$.
$\Rightarrow \max$. (the values of $x$ at which $f$ is discontinuous)
$=\max .(1,2)=2 $
$=f(1) $
$\Rightarrow k=1$
$ \min \cdot\left(\displaystyle\lim _{x \rightarrow 1^{-}} f(x), \displaystyle\lim _{x \rightarrow 1^{+}} f(x)\right) =\min \cdot\left(\frac{1}{2}, 2\right) $
$=\frac{1}{2}$
$ \Rightarrow m=1, p=2 $
$ \Rightarrow p^{2}-k^{2} m^{2}=4-(1)^{2}(1)^{2}=3 $