Question:
Consider the function : \(f(x)=[x]+|1-x|,\ -1\le x\le3\) where \([x]\) is the greatest integer function. Statement 1 : \(f\) is not continuous at \(x=0,1,2\) and \(3\) Statement 2 : \(f(x)=\begin{cases}-x, & -1\le x\lt0\\ 1-x, & 0\le x\lt1\\ 1+x, & 1\le x\lt2\\ 2+x, & 2\le x\le3\end{cases}\)
Consider the function : \(f(x)=[x]+|1-x|,\ -1\le x\le3\) where \([x]\) is the greatest integer function. Statement 1 : \(f\) is not continuous at \(x=0,1,2\) and \(3\) Statement 2 : \(f(x)=\begin{cases}-x, & -1\le x\lt0\\ 1-x, & 0\le x\lt1\\ 1+x, & 1\le x\lt2\\ 2+x, & 2\le x\le3\end{cases}\)
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Work out \(f\) on each interval yourself and compare with the piecewise form given in Statement 2; then test the left and right limits at 0, 1, 2, 3.
Updated On: Nov 4, 2026
- Statement 1 is true; Statement 2 is false,
- Statement 1 is true; Statement 2 is true; Statement 2 is not correct explanation for Statement 1.
- Statement 1 is true; Statement 2 is true; Statement It is a correct explanation for Statement 1.
- Statement 1 is false; Statement 2 is true.
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The Correct Option is A
Solution and Explanation
On \([1,2)\): \([x]=1\) and \(|1-x|=x-1\), so \(f=x\), not \(1+x\). On \([2,3)\): \([x]=2\), \(|1-x|=x-1\), so \(f=x+1\), not \(2+x\); and \(f(3)=3+2=5\). So the piecewise form in Statement 2 is wrong: Statement 2 is false.
Correct form: \(f=-x\) on \([-1,0)\), \(1-x\) on \([0,1)\), \(x\) on \([1,2)\), \(x+1\) on \([2,3)\), \(f(3)=5\).
At \(x=0\): left limit \(0\), \(f(0)=1\): discontinuous. At \(x=1\): left limit \(0\), \(f(1)=1\): discontinuous. At \(x=2\): left limit \(2\), \(f(2)=3\): discontinuous. At \(x=3\): left limit \(4\), \(f(3)=5\): discontinuous.
So Statement 1 is true.
Answer: (1) Statement 1 is true; Statement 2 is false,
Correct form: \(f=-x\) on \([-1,0)\), \(1-x\) on \([0,1)\), \(x\) on \([1,2)\), \(x+1\) on \([2,3)\), \(f(3)=5\).
At \(x=0\): left limit \(0\), \(f(0)=1\): discontinuous. At \(x=1\): left limit \(0\), \(f(1)=1\): discontinuous. At \(x=2\): left limit \(2\), \(f(2)=3\): discontinuous. At \(x=3\): left limit \(4\), \(f(3)=5\): discontinuous.
So Statement 1 is true.
Answer: (1) Statement 1 is true; Statement 2 is false,
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