Unformatted text preview: 316 CHAPTER 3. REALVALUED FUNCTIONS: DIFFERENTIATION
Deﬁnition 3.6.1. Suppose S is a set of real numbers.
1. α ∈ R is a lower bound for S if
α ≤ s for every s ∈ S.
The set S is bounded below if there exists a lower bound for S .
2. β ∈ R is an upper bound for S if
s ≤ β for every s ∈ S.
The set S is bounded above if there exists an upper bound for S .
3. A set of real numbers is bounded if it is bounded below and bounded
above.
4. If S is bounded below, there exists a unique lower bound A for S such
that every lower bound α for S satisﬁes α ≤ A; it is called the
inﬁmum of S , and denoted inf S .
A lower bound α for S equals inf S precisely if there exists a sequence
{si } of elements of S with si → α.
5. If S is bounded above, there exists a unique upper bound B for S
such that every upper bound β for S satisﬁes β ≥ B ; it is called the
supremum of S , and denoted sup S .
An upper bound β for S equals sup S precisely if there exists a
sequence {si } of elements of S with si → β .
6. A lower (resp. upper) bound for S is the minimum (resp.
maximum) of S if it belongs to S . When it exists, the minimum
(resp. maximum) of S is also its inﬁmum (resp. supremum).
These notions can be applied to the image, or set of values taken on by a
realvalued function on a set of points in R2 or R3 (we shall state these for
R3 ; the twodimensional analogues are essentially the same):
Deﬁnition 3.6.2. Suppose f : R3 → R is a realvalued function with domain
dom(f ) ⊂ R3 , and let S ⊂ dom(f ) be any subset of the domain of f . The
image of S under f is the set of values taken on by f among the points of
S:
f (S ) := {f (s)  s ∈ S }. ...
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 Fall '08
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 Calculus, Set Theory, Real Numbers, Supremum, Order theory

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