Continuity & Discontinuity
Increasing & Decreasing
Of Functions
Objective
• SWBAT:
– Identify whether a function is continuous or
discontinuous
– Identify the types of discontinuity
– Identify when a function is increasing, decreasing,
or constant with the intervals respectively.
Definition of Continuity
A function is continuous on
an open interval (a, b) if it is continuous
on each point in the interval. A function
that is continuous on the entire real line
is everywhere continuous.
f(x) is continuous on (-3,2)
A function is continuous if you
can draw it in one motion
without picking up your pencil.
Removable Discontinuities:
(You can fill the hole.)
Nonremovable Discontinuities:
jump
infinite
“Discussing Continuity”
• Continuous or discontinuous?
• If discontinuous
– Removable or nonremovable discontinuity?
– At what x-value is the discontinuity?
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Continuity by Function Type
• Polynomials are everywhere continuous
• Sine and Cosine are everywhere continuous
• Rational functions and other trig functions are continuous except at xvalues where their denominators equal zero.
– “Removable” discontinuity if factoring and canceling “removes” the zero in the
denominator
– “Non-removable” otherwise. (Recall that vertical asymptotes occur where
numerator is nonzero and the denominator is zero.)
• Root functions are continuous, except at x-values that would result in a
negative value under an even root
• For piecewise functions, find the f(x) values at the x-value separating the
regions of the function.
– If the f(x) values are equal, the function is continuous.
– Otherwise, there is a (non-removable) discontinuity at this point.
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Increasing and Decreasing
Functions
Definitions
• Given function f defined on an interval
– For any two numbers x1 and x2 on the interval
• Increasing function
– f(x1) < f(x2) when x1 < x2
• Decreasing function
– f(x1) > f(x2) when x1< x2
X1
X2
X1
X2
f(x)
• Constant Function
– f(x1) = f(x2) when x1< x2
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Check These Functions
• By graphing on calculator, determine the
intervals where these functions are
– Increasing
– Decreasing
f ( x) 
2
x  x  4x  2
3
2
3
y  2 x  5
f ( x) 
x3
x4
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Notes Over 2.3
Increasing and Decreasing Functions
Describe the increasing and decreasing behavior.
8.
The function is
decreasing on the
interval  ,  1
increasing on the
interval  1, 0 
decreasing on the
interval 0, 1
increasing on the
interval
1,  
Decreasing on(-∞, -1) U (0,1)
Increasing on (-1,0) U (1,∞)
Using compound Interval
Notation is More Effective
Notes Over 2.3
Increasing and Decreasing Functions
Describe the increasing and decreasing behavior.
9.
The function is
increasing on the
interval  4,  1

constant on the
interval
 1, 2 

decreasing on the
interval 2, 5
Applications
• Digitari, the great video game
manufacturer determines its
cost and revenue functions
– C(x) = 4.8x - .0004x2 0 ≤ x ≤ 2250
– R(x) = 8.4x - .002x2 0 ≤ x ≤ 2250
• Determine the interval(s) on which the profit
function is increasing
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