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 (integer)
8
Solution
$x \in ( - 10,10)$<br><br>$\Rightarrow$ ${x \over 2} \in ( - 5,5) \to 9$ integers<br><br>check continuity at x = 0<br><br>$$\left. {\matrix{
f & {(0) = } & 0 \cr
f & {({0^ + }) = } & 0 \cr
f & {({0^ - }) = } & 0 \cr
} } \right\}continuous\,at\,x = 0$$<br><br>function will be discontinuous when<br><br>${x \over 2} = \pm 4, \pm 3, \pm 2, \pm 1$
<br><br>For example checking continuity at x = 4<br><br>$$\left. {\matrix{
f & {(4) = } & 4 \cr
f & {({4^ + }) = } & 4 \cr
f & {({4^ - }) = } & 3 \cr
} } \right\}discontinuous\,at\,x = 4$$
<br><br>$\therefore$ 8 points of discontinuity.
About this question
Subject: Mathematics · Chapter: Limits, Continuity and Differentiability · Topic: Limits and Standard Results
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