JEE MAIN - Mathematics (2020 - 5th September Morning Slot - No. 5)
Let $$f(x) = x.\left[ {{x \over 2}} \right]$$, for -10< x < 10, where [t] denotes the greatest integer function. Then the number of points of discontinuity of f is equal to _____.
Answer
8
Explanation
$$x \in ( - 10,10)$$
$$ \Rightarrow $$ $${x \over 2} \in ( - 5,5) \to 9$$ integers
check continuity at x = 0
$$\left. {\matrix{ f & {(0) = } & 0 \cr f & {({0^ + }) = } & 0 \cr f & {({0^ - }) = } & 0 \cr } } \right\}continuous\,at\,x = 0$$
function will be discontinuous when
$${x \over 2} = \pm 4, \pm 3, \pm 2, \pm 1$$
For example checking continuity at x = 4
$$\left. {\matrix{ f & {(4) = } & 4 \cr f & {({4^ + }) = } & 4 \cr f & {({4^ - }) = } & 3 \cr } } \right\}discontinuous\,at\,x = 4$$
$$ \therefore $$ 8 points of discontinuity.
$$ \Rightarrow $$ $${x \over 2} \in ( - 5,5) \to 9$$ integers
check continuity at x = 0
$$\left. {\matrix{ f & {(0) = } & 0 \cr f & {({0^ + }) = } & 0 \cr f & {({0^ - }) = } & 0 \cr } } \right\}continuous\,at\,x = 0$$
function will be discontinuous when
$${x \over 2} = \pm 4, \pm 3, \pm 2, \pm 1$$
For example checking continuity at x = 4
$$\left. {\matrix{ f & {(4) = } & 4 \cr f & {({4^ + }) = } & 4 \cr f & {({4^ - }) = } & 3 \cr } } \right\}discontinuous\,at\,x = 4$$
$$ \therefore $$ 8 points of discontinuity.
Comments (0)
