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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
MCQ
If $f (x)=|x|+|x-1|$, then
  • A
    $f (x)$ is continuous at $x=0$ only
  • B
    $f (x)$ is continuous at $x=1$ only
  • ✓
    $f (x)$ is continuous at both $x=0$ and $x=1$
  • D
    $f (x)$ is discontinuous
✓

Answer

Correct option: C.
$f (x)$ is continuous at both $x=0$ and $x=1$
(C)
Given, $f (x)=|x|+|x-1|$
$\therefore f (x)=\left\{\begin{array}{ll}-x-(x-1), & \text { if } x<0 \\ x-(x-1), & \text { if } 0 \leq x<1 \\ x+(x-1), & \text { if } x \geq 1\end{array}\right.$
$\therefore f (x)=\left\{\begin{array}{ll}-2 x+1, & \text { if } x<0 \\ 1, & \text { if } 0 \leq x<1 \\ 2 x-1, & \text { if } x \geq 1\end{array}\right.$
$\lim _{x \rightarrow 0^{-}} f (x)=\lim _{x \rightarrow 0}(-2 x+1)=1$
$\lim _{x \rightarrow 0^{+}} f (x)=\lim _{x \rightarrow 0} 1=1$
$f(0)=1$
$\therefore \lim _{x \rightarrow 0^{-}} f (x)=\lim _{x \rightarrow 0^{+}} f (x)= f (0)$
$\therefore f (x)$ is continuous at $x=0$.
$\lim _{x \rightarrow 1^{-}} f (x)=\lim _{x \rightarrow 1} 1=1$
$\lim _{x \rightarrow 1^{+}} f (x)=\lim _{x \rightarrow 1}(2 x-1)=1$
$f(1)=2(1)-1=1$
$\therefore \lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)=f(1)$
$\therefore f (x)$ is continuous at $x=1$.

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