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Let $\mathrm{f(x)=|x|+|x-1|}$ then
Option: 1
f(x) is continuous at x=0, as well as at x=1

Option: 2
f(x) is continuous at x=0, but not at x=1

Option: 3
f(x) is continuous at x=1, but not at x=0.

Option: 4
none of these.

Answers (1)

We have, $\mathrm{f(x)=|x|+|x-1|}$
$\mathrm{ =\left\{\begin{array}{rr} -2 x+1, & x<0 \\ x-x+1, & 0 \leq x<1 \\ x+x-1, & x \geq 1 \end{array}=\left\{\begin{array}{rr} -2 x+1, & x<0 \\ 1 & 0 \leq x<1 \\ 2 x-1, & x \geq 1 \end{array}\right.\right.}$
Clearly , $\mathrm{ \lim _{x \rightarrow 0^{-}} f(x)=1, \lim _{x \rightarrow 0^{+}} f(x)=1, \quad \lim _{x \rightarrow 1^{-}} f(x)=1 \quad}$

and

$\mathrm{ \lim _{x \rightarrow 1^{+}} f(x)=1}$ .

So f(x) is continuous at x=0,1.

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Kshitij

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