Math 137 Week 4 Notes – Dr. Paula Smith
I.
Continuity
A. Definitions: Suppose f is de
ﬁ
ned on [a, b].
1. f is
continuous
at x = c
∈
(a, b) if lim
x
→
c
f(x) = f(c).
a)
f(c) must exist
b)
lim
x
→
c
f(x) must exist
c)
they must equal each other
2. f is
continuous from the right
at x = a if lim
x
→
a+
f(x) = f(a).
3.
f is
continuous from the left
at x = b if lim
x
→
b
−
f(x) = f(b).
4.
f is continuous on (a, b) if f is continuous at every point c
∈
(a, b).
5.
f is continuous on [a, b] if f is continuous at every point c
∈
(a, b) and also at continuous from
the right at a and continuous from the left at b.
B. Continuity Laws. Suppose f and g are continuous on interval I.
1.
Sum Law: f ± g is continuous on I.
2.
Product Law: f · g is continuous on I.
3.
Quotient Law: f
is continuous on the subset of I where g
≠
0.
4.
Composition Law: If f is continuous on Ran g
⊆
I, then f(g(x)) is continuous on the subset of
I where the function is de
ﬁ
ned.
C.
Remarks.
1. “Basic” functions such as polynomials, x, a
x
, log
a
x, x
p
, trig functions, arcsin x, arctan x are
continuous at every point in their domains.
This is not necessarily the same as continuous on the
realnumber line.
2.
A function has a “removable” discontinuity at c if lim
x
→
c
f(x) exists.
A function has a
“jump” discontinuity at c if lim
x
→
c
f(x) does not exist.
Geometrically, f is continuous at x = c if
the graph y = f(x) has no break at x = c.
3.
If f is continuous at x = c, then we can interchange the order of f and lim symbol: lim f(x)=
f(lim x) = f(c) [“Substitution Law”]
II.
Intermediate Value Theorem (IVT)
A. IVT
. Suppose f is continuous on [a, b] and f(a)
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 Spring '08
 SPEZIALE
 Math, Continuity, Derivative, Continuous function, Dr. Paula Smith, 137 Week

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