MCV4U – CALCULUS AND VECTORS
Chapter 1.6 – Continuity
Learning Goals – By the end of this section I will know:
-
The mathematical definition of continuity
The different ways a function can have a discontinuity.
How to use limits to identify the presence of a discontinuity.
How to use the results of limits to characterize the type of discontinuity.
CHAPTER 1.6 – CONTINUITY
DEFINITION OF CONTINUITY
 An informal definition of continuity is being able to sketch a graph without having to
lift our pencil, i.e. there are no gaps, holes, or breaks. Of course, we need a
mathematical definition so that we may test or investigate the continuity of a function
at specific values.
A function 𝑦
𝑓 𝑥 is continuous at a number 𝑥
lim 𝑓 𝑥
→
𝑎 if:
𝑓 𝑎
 This definition of continuity gives rise to three specific criteria that must all be satisfied
in order for a function to be continuous at 𝑥 𝑎:
1. 𝑓 𝑎 exists i. e. the function exists at 𝑥
2. lim 𝑓 𝑥 exists i. e. the limit exists at 𝑥
→
3. lim 𝑓 𝑥
→
Ch. 1.6 - Continuity
𝑎
𝑎
𝑓 𝑎 exists i. e. the function and the limit have the same value at 𝑥
𝑎
2
DEFINITION OF CONTINUITY AND TYPES OF DISCONTINUITIES
 If a function 𝑦
𝑓 𝑥 is not continuous at a point 𝑥
𝑎 then we say:
-
The function 𝑦
𝑓 𝑥 is discontinuous at 𝑥
-
The function 𝑦
𝑓 𝑥 has a discontinuity at 𝑥
𝑎 or
𝑎.
 There are three types of discontinuities we will examine:
1.
Removable or point discontinuity.
2.
Jump discontinuity.
3.
Infinite discontinuity (i.e. a break due to an
asymptote).
Continuous for all
values on domain
Point discontinuity
at 𝑥 1
Jump discontinuity
at 𝑥 1
Infinite discontinuity
at 𝑥 1
 We can use limits to investigate continuity at points of interest on a
function. Each type of discontinuity has a specific type of behaviour
which is revealed when investigate the function and its limits at points
of interest.
Ch. 1.6 - Continuity
3
REMOVABLE OR POINT DISCONTINUITY
 A function 𝑦
Case 1:
𝑓 𝑥 has a removable or point discontinuity at 𝑥
1. lim 𝑓 𝑥
→
𝑎:
exists
2. 𝑓 𝑎 does not exist
Case 2:
1. lim 𝑓 𝑥
exists
2. lim 𝑓 𝑥
𝑓 𝑎
→
→
Ch. 1.6 - Continuity
4
JUMP DISCONTINUITY
 A function 𝑦
𝑓 𝑥 has a jump discontinuity at 𝑥
1. lim 𝑓 𝑥
𝐿
2. lim 𝑓 𝑥
𝐿
→
→
3. 𝐿
Ch. 1.6 - Continuity
𝑎:
𝐿
5
INFINITE DISCONTINUITY
 A function 𝑦
𝑓 𝑥 has a infinite discontinuity at 𝑥
1. lim 𝑓 𝑥
∞ or
∞
2. lim 𝑓 𝑥
∞ or
∞
→
→
Ch. 1.6 - Continuity
𝑎:
6
EXAMPLE
For the function 𝑦 𝑓 𝑥 shown, analyze the continuity of this function at the following points (i.e. determine if the function is
continuous, and if it has a discontinuity, characterize the type of discontinuity).
a)
𝑥
3
b)
𝑥
2
c)
𝑥
1
d)
𝑥
0
e)
𝑥
3
Ch. 1.6 - Continuity
7
EXAMPLE
Analyze the continuity of the following function:
Ch. 1.6 - Continuity
𝑓 𝑥
𝑥
𝑥
𝑥
1
𝑥
0
𝑥
0
𝑥
1
1
8