Question

Assertion :Consider the function $f\left(x\right)=[x-1]+|x-2|$ where [.] denotes the greatest integer function.
Statement 1: $f\left(x\right)$ is discontinuous at $x=2$. Reason: Statement 2: $f\left(x\right)$ is not derivable at $x=2$.

• A. Statement 1 is true, Statement 2 is true and Statement 2 is correct explanation for Statement 1.
• B. Statement 1 is true, Statement 2 is true and Statement 2 is NOT the correct explanation for Statement 1.
• C. Statement 1 is true, Statement 2 is false.
• D. Statement 1 is false, Statement 2 is true.

Answer 1

The correct option is D Statement 1 is false, Statement 2 is true.

For $x\to 2^{-}$, $[x-1]=0$, and $|x-2|=2-x$.

For $x\to 2^{+}$, $[x-1]=1$, and $|x-2|=x-2$.

At $x=2$, $[x-1]=1$, and $|x-2|=0$.

So, ${lim}_{x\to 2^{-}}f\left(x\right)=2-x$

${lim}_{x\to 2^{+}}f\left(x\right)=1+x-2=x-1$.

${lim}_{x\to 2^{-}}f\left(x\right)={lim}_{x\to 2^{-}}(2-x)=1{lim}_{x\to 2^{+}}f\left(x\right)={lim}_{x\to 2^{+}}(1+x-2)=1$, which is equal to the function at $x=2$.

Therefore, the function is continuous. However, because it changes its definition at this point, it is not differentiable.

Statement 1 is false, Statement 2 is true.