(2_a)

Step 1:

The function is and .

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 function is .

Substitute in the function.

.

is defined.

Step 2:

Condition (2): exists.

.

exists.

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 .

(2_b)

Step 1:

The function is and .

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 function is .

Substitute in the function.

.

is defined.

Step 2:

Condition (2): exists.

.

exists.

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 .

(2_c)

Step 1:

The function is and .

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 function is .

Substitute in the function.

is undefined.

The condition (1) of continuity is not satisfied, hence the function is discontinuous.

The function is discontinuous at .

Solution :

The function is discontinuous at .

(3_a)

Step 1:

The function is and .

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.

If then .

Substitute in above function.

is defined.

Step 2:

Condition (2): exists.

.

exists.

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 .

(3_b)

Step 1:

The function is and .

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.

If then .

Substitute in above function.

.

is defined.

Step 2:

Condition (2): exists.

Left hand limit :

If then .

.

Right hand limit :

If then .

.

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

does not exists.

Here limit does not exist at

The condition (2) of continuity are not satisfied, hence the function is discontinuous.

The function is discontinuous at .

Solution :

The function is discontinuous at .

(4_a)

Step 1:

The function is and .

Simplify the function.

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 function is .

Substitute in the function.

is undefined.

The condition (1) of continuity is not satisfied, hence the function is discontinuous.

The function is discontinuous at .

Solution :

The function is discontinuous at . 

(4_b)

Step 1:

The function is and .

Simplify the function.

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 function is .

Substitute in the function.

.

 is defined.

Step 2:

Condition (2): exists.

.

exists.

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 .

(4_c)

Step 1:

The function is and .

Simplify the function.

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 function is .

Substitute in the function.

is undefined.

The condition (1) of continuity is not satisfied, hence the function is discontinuous.

The function is discontinuous at .

Solution :

The function is discontinuous at .