At which points the functionf(x) = x [x] , where[.] is greatest i… - Vidyadip

MCQ

At which points the function$f(x) = \frac{x}{{[x]}}$, where$[.]$ is greatest integer function, is discontinuous

A
Only positive integers
✓
All positive and negative integers and $(0, 1)$
C
All rational numbers
D
None of these

✓

Answer

Correct option: B.

All positive and negative integers and $(0, 1)$

b
(b) $(i)$ When $0 \le x < 1$

$f(x)$ doesn't exist as $[x] = 0$ here.

$(ii)$ Also $\mathop {\lim }\limits_{x \to 1 + } f(x)$ and $\mathop {\lim }\limits_{x \to 1 - } f(x)$ does not exist.

Hence $f(x)$ is discontinuous at all integers and also in $(0, 1).$

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