Engineering Calculus Notes 328

Engineering Calculus Notes 328 - 316 CHAPTER 3 REAL-VALUED...

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Unformatted text preview: 316 CHAPTER 3. REAL-VALUED 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 real-valued function on a set of points in R2 or R3 (we shall state these for R3 ; the two-dimensional analogues are essentially the same): Deﬁnition 3.6.2. Suppose f : R3 → R is a real-valued 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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