1. Section 5 – Continuity and End Behavior
Chapter 3
Section 5 – Continuity and End Behavior

2. Discontinuity
Most of the graphs that we’ve seen so far have been smooth, continuous functions.
When a function is discontinuous, it is because it looks “broken” or untraceable without lifting your pencil.

3. Types of Discontinuity
Infinite Dscnty
Jump Dscnty
Point Dscnty

4. Determining Continuity at a Given Point
The Continuity Test:
To see if a function is continuous at a given point: (x=a)
Does f(a) exist??? Plug the given value of x into the function and see if it is solvable. [cannot have zero denominator, cannot have −1 imaginary numbers]
X=1
Exsists!
X=2 is undefined

5. Determining Continuity at a Given Point
The Continuity Test:
Does f(x) approach the same value from both sides? [as x increases toward a or decreases toward a, does it approach the same thing?]
The actual y-value that f(x) approaches from both side IS f(x) (rather than approaching a point-discontinuity)
Discontinuities will usually mean x in the denominator of a fraction and/or x inside an even numbered root [like: 𝟏 𝒙 or 𝟒 𝒙−𝟐 ]

6. Determining Continuity at a Given Point
EX 1: Determine whether f(x)=3x2+7 is continuous at x=1.
EX 2: Determine whether
g(x)=(x-2)/(x2-4) is continuous at x=2
EX 3: Determine whether f(x)= 1/x,x>1
x, x≤1
is continuous at x=1

7. End Behavior
Another tool for analyzing functions is to see the end behavior of a function, which is to say as x approaches infinity and negative infinity.

8. End Behavior for Polynomial Functions

9. Assignment
Chapter 3, Section 5
pgs #6,8,12-28E,33