(11)

Step 1 :

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.

Observe the Graph :

At point

.

is defined.

Step 2:

Condition (2): exists.

Observe the graph.

Left hand limit :

As x tends to from left side, approaches to . .

Right hand limit :

As x tends to from right side, approaches to .

.

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

exist.

Step 3:

Condition (3): .

and .

The three conditions of continuity are not satisfied, hence the function is discontinuous.

The function is discontinuous at .

Solution :

The function is discontinuous at .

 (12)

Step 1 :

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.

Observe the Graph :

At point .

.

is defined.

Step 2:

Condition (2): exists.

Observe the graph.

Left hand limit :

As x tends to from left side, approaches to . .

Right hand limit :

As x tends to from right side, approaches to .

.

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

.

exist.

Step 3:

Condition (3): .

and .

.

The three conditions of continuity are satisfied, hence the function is continuous.

The function is continuous at .

Solution :

The function is continuous at .

(13)

Step 1:

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.

Observe the Graph :

At point .

is undefined.

The conditions of continuity are not satisfied, hence the function is discontinuous.

The function is discontinuous at .

Solution :

The function is discontinuous at .