Let a,b∈R with a<b, and let f:[a,b]→R. Suppose that f has a local extremum at an interior point c∈(a,b) and that f is differentiable at c. Then f′(c)=0.
Let a,b∈R with a<b, and let f:[a,b]→R be continuous at every point in [a,b]. Then there exist points xmin,xmax∈[a,b] such that f(xmin)≤f(x)≤f(xmax)for all x∈[a,b].
Let I be an interval in the sense of Interval in the Real Line, let f:I→R, and let c∈I. One says that f has a local extremum at c if either f has a local maximum at c or f has a local minimum at c; that is, there exists δ>0 such that for eve…
Let (an)n=1∞ be a sequence of real numbers, and let L∈R. One says that (an) converges to L if for every ε>0 there exists N∈N such that for all integers n≥N, ∣an−L∣<ε. In that case one writes…
Let f:[a,b]→R, and let P:a=x0<x1<⋯<xn=b be a partition of [a,b]. For each i=1,…,n, let Mi=sup{f(t):t∈[xi−1,xi]},mi=\reftextdef:lower−bound−infimum−c54−2026ainf{f(t):t∈[xi−1,xi]}. The upper sum and lower sum…
Let S⊆R. A number ℓ∈R is called a lower bound of S if ℓ≤s for every s∈S. If S is nonempty and bounded below, then a number m∈R is called the greatest lower bound, or infimum, of S if:…
Let E⊆R and let f:E→R. The function f is said to be uniformly continuous on E if for every ε>0 there exists δ>0 such that for all x,y∈E, if ∣x−y∣<δ, then ∣f(x)−f(y)∣<ε.
Let a,b,c∈R satisfy a≤b≤c, and let f:[a,c]→R be Riemann integrable on [a,c]. Then the restrictions f∣[a,b]:[a,b]→R and f∣[b,c]:[b,c]→R are Riemann integrable on [a,b] and [b,c], respectively, and…
Let a,b∈R with a<b, let f:[a,b]→R, let P=(x0,x1,…,xn) be a partition of [a,b], and let (t1,…,tn) determine a tagged partition of [a,b] relative to P. The corresponding Riemann sum of f is the real number…
Let a,b∈R with a<b, and let P=(x0,x1,…,xn) be a partition of [a,b]. A tagged partition of [a,b] relative to P is a choice of points ti∈[xi−1,xi] for each i=1,…,n.
Let a,b∈R with a<b. A partition of [a,b] is a finite sequence P=(x0,x1,…,xn) of real numbers such that a=x0<x1<⋯<xn=b. The mesh of P is defined by ∣P∣=max1≤i≤n(xi−xi−1).
Let S be a set. A total order on S is a binary relation ≤ on S, equivalently a subset of S×S, and we write x≤y to mean that (x,y)∈≤. The relation ≤ is a total order if the following axioms hold. 1. For every x∈S, one has x≤x. [Reflexivi…
A field is a set F together with two binary operations, written as (x,y)↦x+y [addition] and (x,y)↦x⋅y [multiplication], such that the following axioms hold. 1. For all a,b,c∈F, one has (a+b)+c=a+(b+c). [Associativity of addition] 2. There exists a…
An ordered field F in the sense of Ordered Field is Dedekind complete if every nonempty subset X⊆F that is bounded above has a least upper bound in F, in the sense of Upper Bound and Least Upper Bound.