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Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferentiability1 Mark
Question
The function f(x) = x − [x], where [⋅] denotes the greatest integer function is:
  1. Continuous everywhere.
  2. Continuous at integer points only.
  3. Continuous at non-integer points only.
  4. Differentiable everywhere.
✓

Answer

  1. Continuous at non-integer points only.

Solution:

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,

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 

$=\lim\limits_{\text{x}\rightarrow\text{n}^{+}}\text{f(x)}=\text{x}-\text{n}=\text{x}-\text{nAs},$

$\text{LHL}\neq\text{RHL}$ 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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