Question

Consider the function, $f\left(x\right)=\{\begin{array}{c}x\left\{x\right\}+1,\\ 2-\left\{x\right\},\end{array}\begin{array}{c}0\le x<1\\ 1\le x\le 2\end{array}$ where $\left\{x\right\}$ denotes the fractional part of $x$. Find incorrect statements.

• A. $\underset{x\to 1}{lim}f\left(x\right)$ exist
• B. $f\left(0\right)\ne f\left(B\right)$
• C. $f\left(x\right)$ is continuous in $[0,2]$
• D. $f\left(x\right)$ is continuous in $[1,2]$

Answer 1

Answer: (A)

$\underset{x\to 1}{lim}f\left(x\right)$ exist

$\begin{array}{c}f\left(x\right)=x\left\{x\right\}+1foro\le x<1\\ f\left(x\right)=2-\left\{x\right\}for1\le x\le 2\\ {lim}_{x\to 1-}f\left(x\right)=1\times 0+1=1\\ {lim}_{x\to 1+}=2-0=2\\ {lim}_{x\to 1-}\ne {lim}_{x\to 1+}\\ so{lim}_{x\to 1}f\left(x\right)wo{n}^{\prime }texist\end{array}$