Transcript BCC.01.9 – Continuity and Differentiability of Functions

Unit B - Differentiation
Lesson B.3.2 - Continuity
Calculus - Santowski
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Lesson Objectives
• 1. Define continuity and know the 3 conditions of
•
•
•
•
•
continuity
2. Understand one sided limits
3. Understand the conditions under which a function is
NOT continuous on both open and closed intervals
4. Use algebraic, graphic, & numeric methods to
determine continuity or points of discontinuity in a
function
5. Sketch graphs having various limit and continuity
conditions
6. Apply continuity to application/real world problems
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Fast Five
• In order to effectively work with continuity and
functions, certain algebra and number skills need to
be in place. No GDCs
• 1. Evaluate sin(-)
• 2. Evaluate sin(0.5)
• 3. Evaluate f(0.5) and f(p) if f(x) = e(2x/) (exact
and approx)
• 4. Evaluate f() and f(3) if f(x) = 1/(x - )2 (exact and
approx)
• 5. Simplify g(x) = (x2 - 4)/(x - 2) and sketch y = g(x)
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Explore
• Two functions, f(x) (on the domain of [-,3]) and
g(x) (on xER), are defined as:

sin x
 2x
 
f (x)  e

 1
x   2

x 2  4

g(x)   x  2

3
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  
x   , 
 2 
 
x   ,  
2 
x   ,3 
x  R|x 2
x 2
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Explore
• (1) Sketch f(x) and g(x) on separate graphs. No
•
GDCs
(2) Determine the limits of f(x) at the following
points:




(a) x = -
(b) x = 0.5
(c ) x = 
(d) x = 3
• (3) Determine the limits of g(x) at x = 2
• (4) Which functions are continuous. Why or why
not?
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Explore
Qui ckTi me™ and a
TIFF (Uncompressed) decompressor
are needed to see this pictur e.
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(A) Continuity
• We have just finished introducing you to
and playing around with limits
• One other application of limits in the
analysis of functions is the idea of
continuity
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(A) Continuity
• We can introduce another characteristic of
functions  that of continuity. We can
understand continuity in several ways:
• (1) a continuous process is one that takes place
gradually, smoothly, without interruptions or
abrupt changes
• (2) a function is continuous if you can take your
pencil and can trace over the graph with one
uninterrupted motion
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(A) Conditions for Continuity
• a function, f(x), is continuous at a given number, x = c, if:
•
(i)
• (ii)
f(c) exists;
lim f (x)
x c
exists
f (x)  f (c)
• (iii) lim
xc


• In other words, if I can evaluate a function at a given value of x = c
and if I can determine the value of the limit of the function at x = c
and if we notice that the function value is the same as the limit
value, then the function is continuous at that point.
• A function is continuous over its domain if it is continuous at
each point in its domain.
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(B) Types of Discontinuities
• (I) Jump Discontinuities:
• ex
x  3
f (x)  
2
1 x
x 1
x 1
• Determine the limit and

function values at x = 1.
• We notice our function
values and our limits (LHL
and RHL) "jump" from 4
to 0
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(B) Types of Discontinuities
• (II) Infinite Discontinuities
•

 1
ex. f (x)  x 2

1
x 0
x 0
• determine the limit and

•
function values at x = 0.
The left hand limit and
right hand limits are both
infinite although the
function value is 1
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(B) Types of Discontinuities
• (III) Removable Discontinuities
x 2  x  2

• Ex f (x)   x  2

1

x 2
x 2
• Determine the limit and
function values at x = 2.
• The left hand limit and right
hand limits are equal to 3
although the function value is 1
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(C) Continuity - Examples
• Find all numbers, x = a, for which each function is discontinuous. For
each discontinuity, state which of the three conditions are not satisfied.
x
• (i) f (x) 
2
x  1
 • (iii)
• (iv)

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(ii)
x2  9
f (x) 
x3
4
3
2

2x

3x

x
 x 1 x  2
 2
f (x)  x  2x  3 
x 2

 x 1
x 2  3x 10

x 2
f (x)   x  2

x 2
7
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(C) Continuity - Examples
• A function is defined as follows:
 e x  a

f (x)   x  2
b  e x 3

x  2
2  x  3
x3
• (i) Evaluate limx-2 if a = 1
• (ii)
Evaluate limx3 if b = 1
• (iii) find values for a and b such f(x) is continuous at BOTH x = -2 and
x=3
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(C) Continuity - Examples
• Define a piecewise function as f(x) where
 x  c, x  3

f (x)   2  bx, 3  x  1
x 3  bx, x  1

• (a) Find a relationship between b and c such that f(x) is continuous at
-3. Then give a specific numerical example of values for b and c
• (b) Find value(s) for b such that f(x) is continuous at 1
• (c ) Find values for b and c such that f(x) is continuous on x€R
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(D) Continuity on a Closed
Interval
• But what about a function
•
that is only defined on a
limited domain i.e. [-4,4]?
Consider a function like
f(x) = √(4 - x2)
• What do we do about the
endpoints at x = +2?
• So we must modify our
limit definition of
continuity slightly
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(D) Continuity on a Closed
Interval - One Sided Limits
• We have considered the idea of a limit as an
•
•
investigation of function behaviour at a given x
value
Until now, each of our functions has allowed us
to look at the x value in question from BOTH
sides (xa+ and xa-)
But what if we can’t consider approaching x from
both sides? How do we then find a limit?
• Solution  introduce the idea of a one sided limit
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(D) Continuity on a Closed
Interval - One Sided Limits
• Some functions have
• For example:
certain domain
restrictions along the
lines of x > a (like log
fcns, even root fcns)
lim ln(2  x) 
x 2
1

x 1
lim
x 1
• So in these cases, we
work with one sided
limits
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
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lim
x 0.5
4
1 2x 
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(D) Continuity on a Closed
Interval - One Sided Limits
• A fcn will be defined
as being continuous
from the right if
lim f (x)  lim f (x)  f (c)
xc
xc
• A fcn will be
Qui ckTi me™ and a
TIFF (Uncompressed) decompressor
are needed to see this pictur e.
considered as being
continuous from the
left if
lim f (x)  lim f (x)  f (c)
xc
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xc
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(E) Importance of Continuities
• Continuity is important for three reasons:
• (1) Intermediate Value Theorem (IVT)
• (2) Extreme Value Theorem (EVT)
• (3) differentiability of a function at a point for now, the basic idea of being able to
draw a tangent line to a function at a given
point for x
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(F) Internet Links for
Continuity
• Calculus I (Math 2413) - Limits - Continuity from Paul
Dawkins
• A great discussion plus graphs from Stefan Waner at
Hofstra U  Continuity and Differentiability  then do
the Continuity and Differentiability Exercises on this site
• Here are a couple of links to Visual Calculus from UTK
 General discussion plus examples and explanations: Continuous
Functions
 Quiz to take on continuous functions: Continuity quiz
 And a second, different type of quiz: Visual Calculus - Drill Continuity of Piecewise Defined Functions
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(G) Homework
• Textbook, S3.2, p164-166
•
•
•
•
(1) Q4,5,6 work with graphs
(2) Q7,13,15,17 work with algebra
(3) Q19,23,25,27 work with piecewise fcns
(4) Q33,37,39 applications
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(H) ECO (extra credit
opportunity)
• Research the Intermediate Value Theorem
• Tell me what it is, why it is important, what
continuity has to do with it and be able to
use it
• MAX 2 page hand written report (plus
graphs plus algebra) + 2 Q quiz
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