This document provides an overview of continuity of functions including: 1. The three conditions for a function f(x) to be continuous at a point x=c. 2. Examples of determining if functions are continuous or discontinuous at given points. 3. Definitions of a function being continuous at endpoints and across an entire interval. 4. Theorems regarding continuity of algebraic combinations of functions and polynomials. 5. The two types of discontinuity - removable and non-removable - and the concept of continuous extensions. 6. Examples of determining continuous extensions of functions at given points.

1. Applied Calculus
1st
Semester
Lecture -04
Muhammad Rafiq
Assistant Professor
University of Central Punjab
Lahore Pakistan

2. CONTINUITY
Continuity Of a Function at a Point
(Continuity Test)
A function f(x) is continuous at x=c, c f if and only if it meets
the following three conditions:
1. exists.
2. → exist i.e → →
3. → =f(c)
EXAMPLE NO. 1
Discuss the continuity of
f(x)= at x=3

3. Solution:
f(3)=2(3)+1=7
→
→ →
→ →
→ →
→ Does not exist.
is discontinuous at x=3.

4. EXAMPLE NO. 2
Discuss the continuity of f(x) = |x-5|, at x=5.
f(x)=
f(x)=
Solution:
f(5)=5-5=0
→
→ →
→ →

5. → →
→
→
is continuous at x=5
EXAMPLE NO.3
Discuss the continuity of
f(x)=
at x=3.
Solution:
f(3)=6

6. → →
= →
( )( )
( )
= →
→
f(x) is continuous at x=3
Definition:-
A function ‘f’ is continuous at left end-
point x=a of the interval[a,b],if
→ = f(a)

7. Definition:-
A function “f” is continuous at right end
point x=b of the interval [a,b],if
→ = f(b)
Definition:- A function “f” is continuous in an interval [a,b]
if it is continuous for all x [a,b]
Theorem# 1:-
(continuity of algebraic combinations)
If function “f” and “g” are continuous at x=c , the
following functions are also continuous at x=c
(1) f+g & f-g
(2) fg

8. (3) , provided g(c)
(4) kf where ‘k’ is any constant
/
,provided /
is defined on an interval
containing “c” and m,n are integers.
Theorem # 2:-
Every polynomial is continuous at every point of the real
line.
Every rational function is continuous at every point where
its denominator is not zero.
f(x) =
continuous at every point except 1.

9. Theorem # 3:-
If ‘f’ is continuous at “c” and ‘g’ is continuous at f(c)
then gof is continuous at x=c.
Types of discontinuity:-
There are two types of discontinuity .
Non – Removable discontinuity:-
f(x)=
f(0) does not exist.
→ does not exist.

10. Definition:-
If at any point, a function does not have
functional value as well as limiting value then the type of
discontinuity is called non removable discontinuity.
Removable discontinuity :-
If at any point, a function does not have functional value
but it has limiting value at that point then that kind of
discontinuity is called removable discontinuity.
e.g. f(x)= x 2
f(2) does not exist
→ = 4

11. Continuous extension :-
If f(c) does not exist . but → = L then we can
define a new function as
F(x) =
e.g.
F(x) =
Then the function F(x) is continuous at x=c . It is called
continuous extension of f(x) to x=c.

12. Example # 1:-
Show that
f(x) =
has a continuous extension to x=2 . Also find that
extension.
Solution:-
f(2) does not exist.
→ = →
= →
( )( )
( )( )
=

13. =
Thus f(x) has a continuous extension at x=2 which is
given by
F(x) =
Example # 2:-
For what value of ‘a’
F(x) =
is continuous at every “x”.

14. Solution:-
Since f(x) is continuous for every ‘x’ .
It is continuous at x = 3
then
→ = →
→ = →
9-1 = 2a(3)
8 = 6a
a =

15. Example # 3:-
Define g(3) in a way that extends 2
to be continous at x =3.
Solution:
g(x) = (
g(3) does not exist .
→ g(x) = →
→
→
=(3+3)=6

16. Thus g(x) has a continues extension to x=3 which is given
by
g(x) =
Example # 4-
Define f(1) in a way that extends f(s)=( to be
continous at s=1.
Solution:
f(s)=(

17. f(1) does not exists.
→ →
→
→
Thus f(s) has a continuous extension to x=1 which is given by
Example # 5:-
→

18. Solution:
→
Example # 6:-
→

19. Solution:
→
=
=
=0
Exercise 1.5
Q (1-40) Odd Problems