Question

If $f\left(x\right)=\{\begin{array}{c}x,when0<x1/2\\ 1,whenx=1/2\\ 1-x,when1/2<x<1\end{array}$ , then

• A. $f\left(x\right)$ is continuous for all real $x$
• B. $f\left(x\right)$ is discontinuous for all real $x$
• C. $f\left(x\right)$ is continuous only at $x=\frac{1}{2}$
• D. $f\left(x\right)$ is discontinuous only at $x=\frac{1}{2}$

Answer 1

The correct option is C $f\left(x\right)$ is continuous only at $x=\frac{1}{2}$

Let $a$ be any real number

$\underset{x\to a}{lim}f\left(x\right)=\underset{x\to a}{lim}x=a$ $($ when $x\to a$ through rational values$)$

$\underset{x\to a}{lim}f\left(x\right)=\underset{x\to a}{lim}(1-x)=1-a$ $($ when $x\to a$ through irrational values$)$

$\underset{x\to a}{lim}f\left(x\right)$ will exist only when $a=1-a$ or $a=\frac{1}{2}$

Thus, if $x\ne \frac{1}{2},$ then $\underset{x\to 0}{lim}f\left(x\right)$ will not exist and hence $f\left(x\right)$ will be discontinuous at $x=a$

where $a\ne \frac{1}{2}$

Also. $\underset{x\to 0}{lim}f\left(x\right)=\frac{1}{2}$ and $f\left(\frac{1}{2}\right)=\frac{1}{2}$

Hence, $f\left(x\right)$ is continuous at $x=\frac{1}{2}$