Question

$f\left(x\right)=\frac{\left[x\right]+1}{\left\{x\right\}+1}forf:[0,\frac{5}{2})\to (\frac{1}{2},3]$, where [.] represents G.I.F. and {.} represents F.P.F., then

• A. f(x) is injective discontinuous function
• B. f(x) is surjective non-differentiable function
• C. $min\left({lim}_{x\to 1^{-}}f\right(x),{lim}_{x\to 1^{+}}f(x\left)\right)=f\left(1\right)$
• D. number of points of discontinuity = f(1)

Answer 1

Answer: (A), (B), (D)

The correct options are
A

f(x) is injective discontinuous function

B

f(x) is surjective non-differentiable function

D

number of points of discontinuity = f(1)

$f\left(x\right)=\{\begin{array}{ccc}\frac{1}{x+1}& ;& 0\le x<1\\ \frac{2}{x}& ;& 1\le x<2\\ \frac{3}{x-1}& ;& 2\le x<\frac{5}{2}\end{array}$

Clearly, f(x) is discontinuous and bijective function.
${lim}_{x\to 1^{-}}f\left(x\right)=\frac{1}{2},{lim}_{x\to 1^{+}}f\left(x\right)=2$