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

The Correct Option is B

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.