Transcript 1.4 Continuity and One-Sided Limits

1.4 Continuity and One-Sided Limits
Main Ideas
• Determine continuity at a point and continuity on
an open interval.
• Determine one-sided limits and continuity on a
closed interval.
• Use properties of continuity.
• Understand and use the Intermediate Value
Theorem.
Informal Definition of continuity
• Graph with only one pen stroke.
• Predictable
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Continuity
Definition of Continuity
Continuity at a Point:
A function f is continuous at c if the following three
conditions are met.
1. f(c) is defined. (a point exists)
2. Lim f(x) exists. (no gap or jump in the graph)
3. Lim f(x) = f(c). (no hole in the graph)
Continuity on an Open Interval:
A function is continuous on an open interval (a,b) if it is
continuous at each point in the interval. A function
that is continuous an the entire real line (- ∞ , ∞ ) is
everywhere continuous.
Discontinuous
If the function fails one of the conditions for continuity
then it is discontinuous at c and any open interval
that contains c.
There are two types of discontinuities;
Removable (Hole)
If f can be made continuous by appropriately
redefining f(c).
Nonremovable (Asymptote or jump)
When f(c) can not be defined.
Identify if the function continuous? If not label the
discontinuity.
One-sided limits
Limit from the right of c.
lim f(x) = L
– As you approach c from values that are larger than c.
Limit from the left of c. lim f(x) = L
– As you approach c from values that are smaller than c.
Remember
lim f(x) = L exists only
equal to each other!
if the one sided limits are
One-sided limits also let us expand upon the
definition of continuity to include a closed
interval.
Continuity on a closed interval
A function is continuous on a closed interval [a,b]
if it is continuous on the open interval (a,b) and
the function is continuous from the right at the
left-hand endpoint and continuous from the left
at the right-hand endpoint.
What kind of function(s) are continuous over an open
interval?
•
Polynomial
What kind of function(s) are continuous over an open
interval of their domain?
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•
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Rational
Radical
Trigonometric
What kind of function(s) do you need to use the
definition of continuity to determine if they are
continuous?
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Piece-wise
Absolute value
Composite
Intermediate Value Theorem
If f is continuous on the closed interval [a,b] and
k is any number between f(a) and f(b), then
there is at least one number c in [a,b] such
that f(c) = k.
How is the theorem used?
1. It is an existence theorem so it is used in proofs of
many other mathematical concepts.
2. To locate zero’s of a function that are continuous.