Question

Show that the function defined by is discontinuous at all integral point. Here denotes the greatest integer less than or equal to x .

Answer 1

The given function is,

g( x )=x−[ x ]

Let k be any integer, then the function becomes,

g( k )=k−[ k ] =k−k =0

The left hand limit of the function is,

LHL= lim x→ k − f( x ) = lim x→ k − x−[ x ] =k−( k−1 ) =1

The right hand limit of the function is,

RHL= lim x→ k + f( x ) = lim x→ k + x−[ x ] =k−k =0

It can be observed that, LHL≠RHL.

Therefore, the function is discontinuous for all integral points.