Limits and Continuity
1
Intro to Continuity
As we have seen some graphs have holes in
them, some have breaks and some have
other irregularities. We wish to study each
of these oddities.
We will use our
information of limits
to decide if a function
is continuous or has
holes.
2
Continuity
Intuitively, a function is said to be continuous
if we can draw a graph of the function with
one continuous line. I. e. without removing
our pencil from the graph paper.
Definition
A function f is continuous at a point x = c if
1.
f (c) is defined
2.
lim f(x) exists
3.
lim f(x)  f (c)
x c
x c
THIS IS THE DEFINITION OF
CONTINUITY
4
Example
f (x) = x – 1 at x = 2.
a. f (2) = 1
1
x11
b. xlim
2
The limit
exist!
c. f (2)  1  lim x  1
2
x2
Therefore the function is continuous at x = 2.
5
Example
f (x) = (x2 – 9)/(x + 3) at x = -3
a. f (-3) = 0/0
b.
Is undefined!
x2  9
lim

x  3 x  3
-6
-3
The limit exist!
c.
x2  9
lim
 f ( 3)
x  3 x  3
-6
Therefore the function is not
continuous at x = -3.
You can use table on your calculator to verify this.
6
Continuity Properties
If two functions are continuous on the
same interval, then their sum, difference,
product, and quotient are continuous on
the same interval except for values of x
that make the denominator 0.
Every polynomial function is continuous.
Every rational function is continuous
except where the denominator is zero.
7
Continuity Summary.
Functions have three types of discontinuity.
Consider 1. Non Removable
Discontinuity at
vertical asymptote.
2. Removable
Discontinuity at
hole.
3. Removable
discontinuity at jump
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Worksheet L.4-1
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