JEE Main 2013 (Online) 25th AprilQuestion 1
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: $f(x)=\begin{cases}-x,&-1\le x<0\\1-x,&0\le x<1\\1+x,&1\le x<2\\2+x,&2\le x\le3\end{cases}$
Limits & Continuity
Solution
$f(x)=-x$ on $[-1,0)$, $1-x$ on $[0,1)$, $x$ on $[1,2)$, $x+1$ on $[2,3)$, $f(3)=5$. Jumps at $0,1,2,3$ ($0\to1$, $0\to1$, $2\to3$, $4\to5$), so Statement 1 is true, while Statement 2 (giving $1+x$ and $2+x$) is false.