Question

The function f (x) = $[ x]^2 - [ x]^2$ (where, [x] is the greatest integer less than or equal to x), is discontinuous at

• A. all integers
• B. all integers except 0 and 1
• C. all integers except 0
• D. all integers except 1

Answer 1

Answer: (B)

all integers except 0 and 1

Answer 2

NOTE All integers are critical point for greatest integer
function.
Case I When x $\in$ I
\hspace22mm f (x) = $[ x]^2 - [ x^2 ] = x^2 - x^2 = 0$
Case II When x $\in$ I
If \hspace22mm 0 < x < 1, then [x] = 0
and \hspace22mm $0 < x^2 < 1, \, then \, [x^2 ] = 0$
Next, if \hspace22mm $1 \le x^2 < 2$
$\Rightarrow$ \hspace22mm $1 \le x < \sqrt 2$
$\Rightarrow$ \hspace22mm [x] = 1 and $[x^2] = 1$
Therefore, \hspace5mm f (x) = $[x]^2 - [ x^2 ] = 0,$ if 1 $\le x < \sqrt 2$
Therefore, \hspace5mm f (x) = 0, if 0 $\le x < \sqrt 2$
This shows that f (x) is continuous at x = 1.
Therefore, f (x) is discontinuous in $(- \infty, 0) \cup [ \sqrt 2, \infty)$
many other points.
Therefore, (b) is the answer.