Question:
The number of points at which the function $f(x)=\frac{1}{x-[x]}$ is not continuous is
(a) 1
(b) 2
(c) 3
(d) none of these
Solution:
The function $f(x)=\frac{1}{x-[x]}$ is discontinuous when $x-[x]=0$.
$x-[x]=0$
$\Rightarrow x=[x]$
$\Rightarrow x$ is an integer
So, $f(x)$ is not continuous for all $x \in Z$.
Thus, the function $f(x)=\frac{1}{x-[x]}$ is not continuous at infinitely many points.
Hence, the correct answer is option (d).
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