# 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

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