$f(x)=\frac{x}{1+[x]}$, is discontinuous in set s, where s is

{[−1, 0) ∪ integers}

$f(x)=\frac{x}{1+n}$, n ≤ x < n + 1

(i) clearly, if n = −1 then f(x) is not defined

i.e. f(x) is not defined ∀ x ∈ [−1, 0)

(ii) if n ≠ −1

$f(x)=-\frac{x}{2}$; −3 ≤ x < −2 (n = −3)

= −x ; −2 ≤ x < −1 (n = −2)

= x ; 0 ≤ x < +1 (n = 0)

= ……….

………….

Clearly, the function is discontinuous at all integral values of x.

From (i) and (ii) f(x) is not continuous in set ‘S’

S = {[−1, 0) ∪ integers}.