Question:

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

Updated On: Jun 23, 2023
  • all integers
  • all integers except 0 and 1
  • all integers except 0
  • all integers except 1
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The Correct Option is B

Solution and Explanation

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.
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Concepts Used:

Functions

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets, mapping from A to B will be a function only when every element in set A has one end only one image in set B.

Kinds of Functions

The different types of functions are - 

One to One Function: When elements of set A have a separate component of set B, we can determine that it is a one-to-one function. Besides, you can also call it injective.

Many to One Function: As the name suggests, here more than two elements in set A are mapped with one element in set B.

Moreover, if it happens that all the elements in set B have pre-images in set A, it is called an onto function or surjective function.

Also, if a function is both one-to-one and onto function, it is known as a bijective. This means, that all the elements of A are mapped with separate elements in B, and A holds a pre-image of elements of B.

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