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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability1 Mark
MCQ
The function $f(x) = x − [x],$ where $[⋅]$ denotes the greatest integer function is :
  • A
    Continuous everywhere.
  • B
    Continuous at integer points only.
  • ✓
    Continuous at non $-$ integer points only.
  • D
    Differentiable everywhere.
✓

Answer

Correct option: C.
Continuous at non $-$ integer points only.
$f(x) = x - x$
Consider $n$ be an integer.
$\text{f(x)}=\text{x}-[\text{x}]=\begin{cases}\text{x}-(\text{n}-1)&\text{n}-1\leq\text{x}<\text{n}\\0&\text{x}=\text{n}\\\text{x}-\text{n}&\text{n}\leq\text{x}<\text{n}+1\end{cases}$
Now,
$\text{LHL}$ at $x = n$
$=\lim\limits_{\text{x}\rightarrow\text{n}^{-}}\text{f(x)}=\text{x}-\text{n}-1=\text{x}-\text{n}+1$
RHL at x = n
$=\lim\limits_{\text{x}\rightarrow\text{n}^{+}}\text{f(x)}=\text{x}-\text{n}=\text{x}-\text{n}$ As,
$\text{LHL}\neq\text{RHL}$ at $x = n$
i.e., given function is not continuous at $n.$
Now, $n$ is any integer.
Therefore, given function is not continuous at integers.
Therefore, given points are continuous at non $-$ integer points only.

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