Question

The number of points at which the function $f\left(x\right)=\frac{1}{x-\left[x\right]}$ is not continuous is
(a) 1
(b) 2
(c) 3
(d) none of these

Answer 1

The function $f\left(x\right)=\frac{1}{x-\left[x\right]}$ is discontinuous when x − [x] = 0.

x − [x] = 0

⇒ x = [x]

⇒ x is an integer

So, f(x) is not continuous for all x ∈ Z.

Thus, the function $f\left(x\right)=\frac{1}{x-\left[x\right]}$ is not continuous at infinitely many points.

Hence, the correct answer is option (d).