Continuity and One-Sided Limits

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Continuity and One-Sided Limits – Part 1
Interactive Notes
I. What does continuous mean?
A function is continuous if there is no ____________________ in the graph (no ___________ or ______________).
A function can be continuous at a certain point or it can be continuous on an interval.
o Continuity at a Point: A function f is continuous at c if the following three conditions are met.
1. _____________________________________________
Example: The following function is not continuous at x  2 because f (2) is not defined.
f ( x) 
x2  4
x2
2. ______________________________________________
Example: The following function is not continuous at x  3 because lim f ( x ) does not exist.
x 3
f ( x) 
x3
x2  9
3. ______________________________________________
Example: The following function is not continuous at x  4 because although lim f ( x ) exists,
x4
lim f ( x)  f (4) .
x 4
 x 2  9, x  4
f ( x)  
x4
3,
o Continuity on an Open Interval: A function is continuous on an open interval if it is continuous at
each point in the interval. A function that is continuous on the entire real line (, ) is
_________________________________________________________.
II. Types of Discontinuities
o There are two types of discontinuities: removable and nonremovable.
o A discontinuity at c is called _____________________________ if f can be made continuous by
appropriately defining or redefining f (c ) . (Removable discontinuities usually appear as
_____________.)
o
 Here are some functions that contain removable discontinuities.
removable discontinuity at _________
removable discontinuity at _________
removable discontinuity at _________
o A discontinuity at c is called ___________________________________ if f can not be made
continuous by appropriately defining or redefining f (c ) . (Nonremovable discontinuities usually
appear as ______________________________.)

Here are some functions that contain nonremovable discontinuities.
nonremovable discontinuity at _________
nonremovable discontinuity at _________
III. How do I detect discontinuities analytically? (Think of _______________________________.)
Find the restricted values for the function. If these values can be eliminated by some technique, then the
discontinuities are _____________________________; however, if these values cannot be eliminated by some
technique, then the discontinuities are __________________________________.
o Examples: Discuss the continuity of the following functions.
o
f ( x) 
1
x3
_______________________________________
_______________________________________
o
x2  x  6
f ( x) 
x2
_______________________________________
_______________________________________
o
f ( x) 
x4
x 2  16
_______________________________________
_______________________________________
o
f ( x)  2 x  3
_______________________________________
_______________________________________
o
2 x  3,
f ( x)  
5,
x0
x0
_______________________________________
_______________________________________
o
2 x  3,
f ( x)  
 x  3,
x0
x0
_______________________________________
_______________________________________
o
2 x  3,
f ( x)  
4,
x0
x0
_______________________________________
_______________________________________
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