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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability1 Mark
MCQ
The function $f(x)=[x]$, where $[x]$ denotes the greatest integer less than or equal to $x$, is continuous at
  • A
    $x=1$
  • B
    $x=1.5$
  • C
    $x=-2$
  • D
    $x=4$
✓

Answer

Let $x=1.5$
$\therefore$ L.H.L. $=\operatorname{Lt}_{x \rightarrow 1.5^{-}} f(x)=\underset{h \rightarrow 0}{\operatorname{Lt}}[1.5-h]=1$
and R.H.L. $=\operatorname{Lt}_{x \rightarrow 1.5^{+}} f(x)=\underset{h \rightarrow 0}{\operatorname{Lt}}[1.5+h]=1$
$\because \quad$ L.H.L. = R.H.L.
$\therefore f(x)$ is continuous at $x=1.5$
Also, greatest integer function is discontinuous at all integral values of $x$.

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