f is continuous ⇒
❶ f is integrable
❷ f(x)ᐃx is differentiable
❸ (f(x)ᐃx)ᐁx = f
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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.
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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⸥}
...
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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 ...
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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
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7106
If f is monotone on [a,b] then f is Riemann integrable on [a,b].
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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
.
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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.
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718
Let f be defined and continuous on [a,b]. Then f is uniformly continuous on [a,b].
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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.
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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 ...
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