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

Solution:

Greatest Integer Function is a function that gives the greatest integer less than or equal to the number.

The given function is g(x) = x − [x]

It is evident that g is defined at all integral points.

Let n be an integer.

Then,

g(n) = n − [n] = n − n = 0

The left hand limit of g at x = n is,

lim_x→n^- g(x)

= lim_x→n^- (x − [x])

= lim_x→n^- (x) − lim_x→n^- [x]

= n − (n − 1) = 1

The right hand limit of g at x = n is,

lim_x→n⁺ g(x)

= lim_x→n⁺ (x − [x])

= lim_x→n⁺ (x) − lim_x→n⁺ [x]

= n − n = 0

It is observed that the left and right-hand limit of g at x = n do not coincide.

Therefore, g is not continuous at x = n.

Hence, g is discontinuous at all integral points

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 19

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

Summary:

Hence we have concluded that the function defined by g(x) = x − [x] is discontinuous at all integral points