(CFIVP) Continuous Functions on Intervals Have the
Intermediate Value Property Theorem 1. If f is continuous on [a,b] and f(a) < 0 < f(b), then there is some x in [a,b] such that f(x) = 0.
An argument constructing the binary expansion for one such x
will be given in class. See
http://www.math.uic.edu/ jlewis/mtht430/chap7b.htm(CFCIB) Continuous Functions on Closed Intervals are
Bounded Theorem 2. If f is continuous on [a,b], then f is bounded above on [a,b], that is, there is some number N such that f(x) ≤ N for all x in [a,b].
CFCIMAX) Continuous Functions on Closed Intervals assume a
Maximum Value for the Interval Theorem 3. If f is continuous on [a,b], then there is a number y in [a,b] such that f(y) ≥ f(x) for all x in [a,b]
Consequences
•
If f is continuous on [a,b] and changes
sign, then the equation f(x) = 0 has a root in (a,b).
•
(Intermediate Value Property for Continuous
Functions on Closed Intervals) If f is continuous on
[a,b] and ξ is between f(a) and f(b), then the
equation f(x) = ξ has a root in (a,b).
•
Every nonnegative number ξ has a unique
nonnegative square root, denoted √{ξ}, which satisfies √{ξ} ≥ 0 and (√{ξ})2 = ξ.
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