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STD 12 - 5. continuity and differentiation — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 5. continuity and differentiation1 Mark
MCQ
Find all the points of discontinuity of the function $f$ defined by

$f(x) = \left\{ {\begin{array}{*{20}{c}}
{x + 2,}&{if\,\,x\,\, < \,\,1}\\
{0,}&{if\,\,\,x = 1}\\
{x - 2,}&{if\,\,x\,\, > \,\,1}
\end{array}} \right.$

  • A
    $x=0$
  • B
    $x=-1$
  • ✓
    $x=1$
  • D
    $x=2$
✓

Answer

Correct option: C.
$x=1$
c
As in the previous example we find that $f$ is continuous at all real numbers $x \neq 1 .$ The left hand limit of $f$ at $x=1$ is

$\mathop {\lim }\limits_{x \to {1^ - }} f(x) = \mathop {\lim }\limits_{x \to {1^ - }} (x + 2) = 1 + 2 = 3$

The right hand limit of $f$ at $x=1$ is

$\mathop {\lim }\limits_{x \to {1^ + }} f(x) = \mathop {\lim }\limits_{x \to {1^ + }} (x - 2) = 1 - 2 =  - 1$

Since, the left and right hand limits of $f$ at $x=1$ do not coincide, $f$ is not continuous at $x=1 .$ Hence $x=1$ is the only point of discontinuity of $f .$ The graph of the function is given in the (Fig)

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