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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
Consider $f(x) = [x] + \sqrt {\left\{ X \right\}}$ where $[.]$ denotes greatest integer function and $\{.\}$ denotes fractional part function. Identify the correct statement-
  • A
    $ƒ(x)$ is continuous for $R^+$ only
  • B
    $ƒ(x)$ is continuous for $R^-$ only
  • C
    $ƒ(x)$ is continuous $\forall x \in R -I$ only
  • ✓
    $ƒ(x)$ is continuous $\forall x \in R$
✓

Answer

Correct option: D.
$ƒ(x)$ is continuous $\forall x \in R$
d
$f(\mathrm{x})$ is continuous for all non integers for integers

$f\left(\mathrm{I}^{+}\right)=\mathrm{I}-0=\mathrm{I}$

$f\left(\mathrm{I}^{-}\right)=\mathrm{I}-1+\sqrt{1}=\mathrm{I}$

$\therefore $ $f(\mathrm{x})$ is continuous for integers

$\therefore $ $f(\mathrm{x})$ is continuous $\forall \mathrm{x} \in \mathrm{R}$

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