#### Explain solution RD Sharma class 12 Chapter 8 Continuity exercise 8.1 question 16

Continuous

Hint:

Continuous function must be defined at a point, limit must exist at the point and the value of the function at that point must equal the value of the right hand or left hand limit at that point.

Solution:

Given,

$f(x)=\left(\begin{array}{l} x, i\! f \: 0 \leq x<\frac{1}{2} \\\\ \frac{1}{2}, \text { if } x=\frac{1}{2} \\\\ 1-x, \text { if } \frac{1}{2}

We observe

[LHL at   $x=\frac{1}{2}$ ]

$\lim _{x \rightarrow \frac{1^{-}}{2}} f(x)=\lim _{h \rightarrow 0} f\left(\frac{1}{2}-h\right)=\lim _{h \rightarrow 0}\left(\frac{1}{2}-h\right)=\frac{1}{2}$

[RHL at  $x=\frac{1}{2}$ ]

$\lim _{x \rightarrow \frac{1^{+}}{2}} f(x)=\lim _{h \rightarrow 0} f\left(\frac{1}{2}+h\right)=\lim _{h \rightarrow 0}\left(1-\left(\frac{1}{2}+h\right)\right)=\frac{1}{2}$

Also  $f\left ( \frac{1}{2} \right )=\frac{1}{2}$

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

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