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) .

The function is .

If , then the function is .

.

 is defined at .

The function is .

Left hand limit :

If  closes to  but smaller than , then the denominator is a small negative number.

The function  gets a large negative number.

.

Right hand limit :

If  closes to  but larger than , then the denominator is a small positive number.

The function  gets a large positive number.

.

Left hand limit and right hand limit are not equal, so limit does not exist.

 does not exist.

It does not satisfies the condition of continuity, hence the function is discontinuous.

Therefore  is discontinuous 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 does not exist because the left and right hand limits are not equal.

The function  is discontinuous at .

The function  is discontinuous at .