Geometrically this means that there is NO gap split or missing pt hole for fx at c A pencil could be moved along the graph of fx through c fc WITHOUT lifting it off of the graph The function not only intended to reach a certain height limit but it actually did ID: 644099
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Slide1
ContinuitySlide2
What is Continuity?
Geometrically, this means that there is NO gap, split, or missing pt. (hole) for f(x) at c.
A pencil could be moved along the graph of f(x) through (c, f(c)) WITHOUT lifting it off of the graph.
The function not only intended to reach a certain height (limit) but it actually did:
Limit exists +
Fnc
. Defined = Continuity
Discontinuity occurs when there is a hole in the graph even if the graph doesn’t actually break into 2 different pieces.
Slide3
Formal Definition of Continuity
f(c) exists (c is in the domain of f)
lim
f(x) exists
lim f(x) = f(c)*NOTE: When a fnc. increases and decreases w/o bound around a vertical asymptote (x=c), then the fnc. demonstrates infinite discontinuity. A function is continuous at a point if the limit is the same as the value of the function.Slide4
Examples
1
2
3
4
1
2
This function has
discontinuities
at x=1 and x=2.
It is continuous at x=0 and x=4, because the one-sided limits match the value of the functionSlide5
Types of Discontinuities:
Removable Discontinuities:
Essential Discontinuities:
Jump Infinite Oscillating
(You can fill the hole.)Slide6
Removing a Discontinuity:
has a discontinuity at
x=1.
Write an extended function that is continuous at
x=1.Slide7
Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous.
Also, composites of continuous functions are continuous.
*NOTE: Graphing calculators can make non-continuous functions appear continuous; the calculator “connects the dots” which covers up the discontinuities.Slide8
Evaluate Continuity at the given pt.:
f(x) = 2x+3 at x = -4
f(x) = at x = 2
f(x) = at x = 0