1.4 Continuity f is continuous at a if 1. is defined. 2. exists. 3.
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Transcript of 1.4 Continuity f is continuous at a if 1. is defined. 2. exists. 3.
1.4 Continuity f is continuous at a if
1. is defined.
2. exists.
3.
xfax
lim
af
afxfax
lim
Ex 1: Discontinuous where & why?
*see graph.
1.4 Continuity 3 types of discontinuity:
Removable Infinite Jump
a)
b)
2
22
x
xxxf
xxf
Ex 2: Discontinuous where & why?
c)
d)
2
1
xxf
2 if 1
2 if 1
x
xxxf
Ex 2: Discontinuous where & why?
Functions are continuous at every
number in their domains!
f is continuous on [a,b] if it is continuous on (a, b) and:
Continuity on a Closed Interval
afxfax
lim
bfxfbx
lim
Ex 3: Show that f(x) is continuous on the interval
[1, 1]
211 xxf
Ex 4: Continuous where?
xxf cos1ln
The Intermediate Value Theorem (IVT):
If f is continuous on the interval [a, b] and k is any
number between f(a) & f(b), then there exists a number c in (a, b) such that f(c) = k.
Ex 5: Show that the equation has a root in the
interval [1, 2]
02364 23 xxx
1.4 pg. 78pg. 781 – 5 odds,1 – 5 odds,
7 – 23 EOO,7 – 23 EOO,25 – 31 odds,25 – 31 odds,33 – 53 EOO,33 – 53 EOO,
57, 59, 75, 77, 8557, 59, 75, 77, 8523 Total23 Total