The piecewise function is , .

Definition of continuity :

A function  is continuous at , if  then it should satisfy three conditions :

(1)  is defined.

(2)  exists.

(3) .

Condition (1):

 is defined.

The piecewise function is .

If  then the function is .

.

 is defined at .

Condition (2):

 exists.

Left hand limit :

.

Right hand limit :

.

Left hand limit and right hand limit are equal, limit exist.

.

Condition (3):

.

 and .

Here .

The third condition of continuity is not satisfied, hence the function is not continuous.

The function  is not continuous at .

Graph :

Graph the piecewise function  :

Observe the graph.

As  approaches to  from left hand side,  tends to .

As  approaches to  from right hand side,  tends to .

Limit exist because the left and right hand limits are equal.

.

But .

.

The function  is discontinuous at .

The function  is discontinuous at .