2.6 Continuity
Intermediate Value Theorem (IVT) for Continuous Func
tions
THEOREM 12  IVT for Continuous Functions
A function
y
=
f
(
x
) that is continuous on a closed interval [
a, b
] takes
on every value between
f
(
a
) and
f
(
b
).
In other words, if
y
0
is any value
between
f
(
a
) and
f
(
b
), then
y
0
=
f
(
c
) for some
c
in [
a, b
].
Geometrically, the Intermediate Value Theorem says that any horizontal line
y
=
y
0
crossing the
y
axis between the numbers
f
(
a
) and
f
(
b
) will cross the
curve
y
=
f
(
x
) at least once over the interval [
a, b
].
A Consequence for Graphing: Connectivity
The graph of a function continuous on an interval cannot have any breaks
over the interval. It is
connected
 a single, unbroken curve. No jumps!
”The graph of a continuous function
f
can be sketched over its domain in
one continuous motion without lifting the pencil.”
1
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A Consequence for Root Finding
A solution fo the equation
f
(
x
) = 0 is a
root
of the equation or
zero
of the
function.
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 Spring '08
 FBHinkelmann
 Calculus, Topology, Continuity, Intermediate Value Theorem, Continuous function, Continuity Review

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