# Fundamental program of the calculus by Smithee A.

By Smithee A.

Submitted, in your attention, a collection of calculus notes from a few not likely destiny, overlaying the fundamental components of the speculation of the spinoff, quintessential, and degree at the actual line.

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If G is bounded then L(G) is finite. If G is unbounded, then L(G) can be finite or infinite depending on the sum of the series. 5. Properties of open sets The definition makes it clear what the structure of an open set is. Often the set we wish to study is not presented to us in a way that reveals the structure. How then might we verify that it is open? A point in an open set must belong to some component interval. That feature provides a local property that characterizes open sets. 12. A set G is open if and only if for every x ∈ G there is an open interval (c, d) containing that point and itself entirely contained in G.

V ∗ (∆F, E) ≤ n=1 Proof. If ∞ V ∗ (∆F, En ) = ∞ n=1 there is nothing to prove. Otherwise, let > 0 and choose full covers βi of Ei so that V (∆F, βi ) < V ∗ (∆F, Ei ) + 2−i . ∞ Check that β = i=1 βi [Ei ] is a full cover of E. Consequently ∞ V ∗ (∆F, E) < V (∆F, β) ≤ ∞ V ∗ (∆F, Ei ) + . 7. 5. 11. Let F : R → R and let E be a set of real numbers. 1) ([u,v],w)∈π whenever π is a subpartition, π ⊂ β[E]. Note: If F : [a, b] → R then the definition should be altered to require a Cousin cover β of [a, b].

2. Full covers and Cousin covers A cover is said to be full at a point x0 if it contains all pairs ([u, v], x0 ) with the interval [u, v] small enough. 1. A covering relation β is said to be full at a point x0 if there is a δ > 0 so that β contains all pairs ([u, v], x0 ) for which u ≤ x0 ≤ v and 0 < v − u < δ. 2. A covering relation β is said to be a full cover of a set E provided that β is full at each point of E. 3. A covering relation β is said to be simply a full cover provided that β is full at each point of the real line.