Question

The function $f\left(x\right)-p[x+1]+q[x-1]$, where is the greatest integer function is continuous at x =1, if

• A. $p-q=0$
• B. $p+q=0$
• C. $p=0$
• D. $q=0$

Answer 1

The correct option is B $p+q=0$

$f\left(x\right)-p[x+1]+q[x-1]andf\left(1\right)=p[1+1]+q\left[0\right]=2p$

This functionwill be continuous at x=1, then L $\underset{x\to 1}{lim}f\left(x\right)=R\underset{x\to 1}{lim}f\left(x\right)=f\left(1\right)$

$\Rightarrow \underset{h\to 0}{lim}f(1-h)-\underset{h\to 0}{lim}f(1+h)-f\left(1\right)$

$\Rightarrow \underset{h\to 0}{lim}p[1-h+1]+q[1-h-1]$

$\Rightarrow \underset{h\to 0}{lim}p[1+h+1]+q[1+h-1]-f\left(1\right)$

$\Rightarrow \underset{h\to 0}{lim}p[2-h]+q[-h]-\underset{h\to 0}{lim}p[2+h]+q\left[h\right]=f\left(1\right)$

$\Rightarrow \underset{h\to 0}{lim}p(1-h)+q(-h-1)-\underset{h\to 0}{lim}\left[p\right(1+h)+q(h-1\left)\right]-2p$

$\Rightarrow p-q=2p\Rightarrow p+q=0.$