Question

Question 19: Show that the function defined by g (x) = x – [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.

Class 12 - Math - Continuity and Differentiability

Answer 1

Here g(x) = x-[x]
let f(x) = x and q(x) = [x]
Now f(x) being an identity function is continuous for all real x
and q(x) = [x] bring greatest integer function is discontinuous at every integer value of x .
also f(x) -q(x) = g(x)

Hence g(x) = x-[x] is discontinuous for all integral values of x .

Answer 2

★ CONTINUITY AND DIFFERENTIABILITY ★

[ x ] denotes the greatest integer function ,

In general , if n is an integer , and x is any real number between n and n + 1 ,

Then , n ≤ x < n + 1 , then , [ x ] = n

It's the general preview about greatest integer function

Now , accordingly , f ( x ) is an identity function
which is obviously continuous for all real x values , and q ( x ) = [ x ] is G.I.F. which is obviously discontinuous for every possible integer value of x
Now , f ( x ) - q ( x ) = g ( x )

aslike , g ( x ) = x - [ x ]

f ( x ) = x and g ( x ) = [ x ]

Therefore , g ( x ) = x - [ x ] is discontinuing function ∀ x ∈ I

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