Question

Is th function defined by f(x) = $\{\begin{array}{c}x+5,ifx\le 1\\ x-5,ifx>1\end{array}$ a contionuous functions ?

Answer 1

Here, f(x) = $\{\begin{array}{c}x+5,ifx\le 1\\ x-5,ifx>1\end{array}$

LHL = $\underset{x\to 1^{-}}{lim}$ f(x) = $\underset{x\to 1^{-}}{lim}$ $x+5$

Putting x=1-h as $x\to 1^{-}$ when $h\to 0$

$\therefore$ $\underset{h\to 0}{lim}$ [1-h+5] =$\underset{h\to 0}{lim}$ [6-h] = 6-0=6

RHL = $\underset{x\to 1^{+}}{lim}$ f(x) = $\underset{x\to 1^{+}}{lim}$ (x-5)

Putting x=2+h as $x\to 1^{+}$ when $x\to 0$

$\underset{h\to 0}{lim}$ (1+h-5) = $\underset{h\to 0}{lim}$ (h-4) = 0-4 =- 4

$\therefore$ LHL $\ne$ RHL Thus, f(x) is not continuous at x=1.

Direction (Q, Nos, 14 to 16) Discuss the continuity of the function defined in the question.