The function f(x) = l x, , , if , ,0 x 1 1, , , , if ,1 < x 2 . i… - Vidyadip

MCQ

The function $f(x) = \left\{ \begin{array}{l}x,\,\,{\rm{if \,\,0}} \le x \le {\rm{1}}\\{\rm{1,\,\,}}\,{\rm{ if}}\,1 < x \le 2\end{array} \right.$ is

A
Continuous at all $x$, $0 \le x \le 2$ and differentiable at all $x$, except $1$ in the interval $(0,2)$
B
Continuous and differentiable at all $x$ in $[0,2]$
C
Not continuous at any point in $[0,2]$
D
Not differentiable at any point $[0,2]$

✓

Answer

$f(x) = \left\{ {\begin{array}{*{20}{l}}{x{\rm{ ,}}}&{0 \le x \le 1}\\{1{\rm{ ,}}}&{1 < x \le 2}\end{array}} \right.$
$\mathop {\lim }\limits_{x \to {1^ - }} f(x) = \mathop {\lim }\limits_{h \to 0} f(1 - h)$
$ = \mathop {\lim }\limits_{h \to 0} \,\,(1 - h) = 1$
$\mathop {\lim }\limits_{x \to {1^ + }} f(x) = \mathop {\lim }\limits_{h \to 0} \,f(1 + h) = 1$
Hence function is continuous in $(0, 2).$
Now $\mathop {\lim }\limits_{x \to {0^ + }} f(x) = \mathop {\lim }\limits_{h \to 0} \,(0 + h) = 0 = f(0)$
$\mathop {\lim }\limits_{x \to {2^ - }} f(x) = \mathop {\lim }\limits_{h \to 0} \,(2 - h) = 1 = f(2)$
Hence function is continuous in $[0, 2]$
Clearly, from graph it is not differentiable at $x = 1.$

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