Question

Consider the greatest integer function, define by $f\left(x\right)=\left[x\right],0\le x<2.$ Then

• A. $f$ is derivable at $x=1$
• B. $f$ is not derivable at $x=1$
• C. $f$ is derivable at $x=2$
• D. None of these

Answer 1

The correct option is D None of these

$f\left(x\right)=\left[x\right],0\le x<2Atx=1,f\left(1\right)=1Atx=1^{+},f\left(1^{+}\right)=1Atx=1^{-},f\left(1^{-}\right)=0sincef\left(1^{-}\right)\ne f\left(1\right)\ne f\left(1^{+}\right)f\left(x\right)isnotcontinousatx=1hencenotderivableatx=1Remark:f\left(x\right)=\left[x\right]isnotdifferentiableateveryintegerpoints$