Let f(x) = [x], where [.] is the greatest integer function and g(x) = sin x be two real valued functions over R.

Which of the following statements is correct?

Concept:

1. Greatest Integer Function: Greatest Integer Function [x] indicates an integral part of the real number x which is nearest and smaller integer to x. It is also known as floor of x

• In general, If  Then [x] = n     (n ∈ Integer)
• Means if x lies in [n, n+1) then the Greatest Integer Function of x will be n.

Example:

2. A function f(x) is said to be continuous at a point x = a, in its domain if  exists or its graph is a single unbroken curve.

f(x) is Continuous at x = a ⇔ 

Calculation:

For f(x) = [x]:

LHL = 

LHL ≠ RHL, so f(x) is discontinuous at x = 0

For g(x) = sin x

g (0) = sin (0) = 0

LHL = RHL = g (0), so g(x) is continuous at x = 0

Hence, option (3) is correct.