7105 Riemann's condition

f is continuous ⇒ ❶ f is integrable ❷ f(x)ᐃx is differentiable ❸ (f(x)ᐃx)ᐁx = f

• 7101 Partition
Let [a,b] be a given finite interval. A partition P of [a,b] is a finite set of points {x₀,x₁,x₂,...,xₙ} satisfying a=x₀<x₁<x₂<...<xₙ=b.
9727#9750 SIBLINGS CHILDREN 9750

• 7102 Upper & Lower Sum
ᐥThe upper sum of a function f relative to a partition P is defined by ｢sup(f(x))ƧP[a,b]｣ The lower sum of a function f relative to a partition P is defined by ｢inf(f(x))ƧP[a,b]｣.ᐥ ┄ U(P) = ｢S⸤i⸥(x⸤i⸥−x⸤i−1⸥) Σi=0,n｣ where S⸤i⸥ = sup{f(x):x⸤i−1⸥≤x≤x⸤i⸥} ...
9727#9751 SIBLINGS CHILDREN 9751

• 7103 Riemann integrable
ᐥA function f defined and bounded on [a,b] is Riemann integrable on [a,b] if and only if inf｢sup(f(x))Ƨ[a,b]｣ = sup｢inf(f(x))Ƨ[a,b]｣ . The common value is denoted by f(x)ᐃx｢a,b｣.ᐥ ┄ inf{U:U=｢sup{f(x):x⸤i−1⸥≤x≤x⸤i⸥}·(x⸤i⸥−x⸤i−1⸥) Σi=0,n｣} = sup{L:L=｢inf{f ...
9727#9753 SIBLINGS CHILDREN 9753

• 7105 Riemann's condition
ᐥLet f be defined and bounded on [a,b]. Then f is Riemann integrable on [a,b] if and only if for every ε>0 there exists a partition P of [a,b] such that inf｢sup(f(x))ƧP[a,b]｣ − sup｢inf(f(x))ƧP[a,b]｣ < ε.ᐥ
9727#9754 SIBLINGS CHILDREN 9754

Songs in Easy English

• 7106
If f is monotone on [a,b] then f is Riemann integrable on [a,b].
9727#9160 SIBLINGS CHILDREN 9160

• 7107 Uniform Continuity
A function f defined on the interval [a,b] is uniformly continuous if for any given ε>0 there exist a δ>0 such that for all x,y∈[a,b] |x−y|<δ ⇒ |f(x)−f(y)|<ε .
9727#9454 SIBLINGS CHILDREN 9454

• 7111 Fundamental theorem of calculus
If f is Riemann integrable on [a,b] then f(t)ᐃt｢a,x｣ is continuous on [a,b]. Further more, f is continuous on [a,b] then f(t)ᐃt｢a,x｣ is differentiable on [a,b] and (f(t)ᐃt｢a,x｣)ᐁx=f.
9727#9755 SIBLINGS CHILDREN 9755

• 718
Let f be defined and continuous on [a,b]. Then f is uniformly continuous on [a,b].
9727#9764 SIBLINGS CHILDREN 9764

• Fundamental theorem of calculus
f is continuous at c if and only if for every ε>0 there exists δ>0 such that |x−c|<δ ⇒ |f(x)−f(c)|<ε. ┄ Continuous at c ｢f(x) Ƚx￫c｣ = f(c) ┄ A function f:A￫ℝ is differentiable at c if and only if ｢(f(x)−f(c))/(x−c) Ƚx￫c｣ exists. The value of the limit, c ...
9727#9759 SIBLINGS CHILDREN 9759