Exercise 9.1

Question 1

Test the continuity of the function on at the origin:

Sol :

Given

We observe

(LHL at x= 0)

=-1

(RHL at x = 0)

=1

Hence, is discontinuous at the origin.

Question 2

A function is defined as

Show that is continuous that

Sol :

Given

We observe

(LHL at x = 3)

= 5

(RHL at x= 3)

= 5

Also , f (3) = 5

Hence, is continuous at

Question 3

A function is defined as

Show that is continuous at

Sol :

Given

We observe

(LHL at x = 3)

= 6

(RHL at x = 3)

= 6

Also , f (3) = 6

Hence, is continuous at

Question 4

If

Find whether is continuous at

Sol :

Given

We observe

(LHL at x = 1)

= 2

(RHL at x = 1)

= 2

Also , f (1) = 2

Hence, is continuous at

Question 5

If

Find whether is continuous at

Sol :

Given

If

We observe

(LHL at x = 0)

= 3

(RHL at x = 0)

= 3

Also , f (0) = 1

Hence, is discontinuous at

Question 6

If

Find whether is continuous at

Sol :

Given

If

We observe

(LHL at x = 0)

= 0

(RHL at x = 0)

= 0

Also , f (0) = 1

Hence, is discontinuous at

Question 7

Let

Show that is discontinuous at

Sol :

Given

Let

Consider

(LHL at x = 0)

(RHL at x = 0)

Also , f (0) = 1

Hence, is discontinuous at

Question 8

Show that

is continuous at

Sol :

Given

If

We observe

(LHL at x = 0)

= 0

(RHL at x = 0)

= 0

Also , f (0) = 2

Hence, is continuous at

Question 9

Show that

is discontinuous at

Sol :

We observe

(LHL at x = a)

= -1

(RHL at x = a)

= 1

Hence, is discontinuous at

Question 10

Discuss the continuity of the following functions at the indicated point(s):

(i) at *x= 0*

(ii) at *x= 0*

(iii) at *x= 0*

(iv) at *x= 0*

(v) at

(vi) at *x = 1*

(vii) at *x = 0*

(viii) at *x = 0*