Graphics Reference
In-Depth Information
Unbounded intervals are classified as closed
half-rays [ a , )or( , a ], open half-rays
( a ,
)or(
, a ), or thewholeline
R
.
The Intermediate Value Theorem
qns 14, 15
A continuous function, f :[ a , b ] R takes every
value between f ( a ) and f ( b ).
Theorem
qn 21
The range of a continuous function defined on
an interval is always an interval.
Theorem
qns 22, 23,
24, 25
If a continuous function, f , is defined on an
interval, f has an inverse function if and only if f
is strictly monotonic. In this casetheinvrse
function is continuous and strictly monotonic.
The Maximum—minimum Theorem
qns 31, 32,
34
A continuous real function defined on a closed
interval is bounded, and attains is bounds.
Uniform continuity
Definition
qns 34 — 42
A function f : A R is said to beuniformly
continuous on A when, given 0, there exists
a
such that
x y f ( x )
f ( y )
.
Theorem
qn 43
If a real function is continuous on a closed
interval then it is uniformly continuous on that
interval.
Theorem
qn 47
If a function is uniformly continuous on a dense
subset of a closed interval, including the end
points, then it may be extended to a continuous
function on thewholeintrval.
Historical note
The Intermediate Value Theorem had been assumed from
geometrical perceptions during the eighteenth century as the basis of
work on approximations to roots of equations. There were
mathematicians who regarded it as the essential characterisation of
continuity. (Wesaw how wrong that was in qn 16. Theinadmissibility
of defining continuous functions by the Intermediate Value Theorem
was pointed out by Darboux in 1875.) In 1817 Bolzano insisted that
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