0. Some Useful Mathematical Definitions and Theorems

이번 챕터에서는 미분방정식 공부 전의 준비 운동으로서 주로 리만 적분의 정의와 Fundamental Theorem of Calculus(FToC)을 포함한 수학 이론들에 대해 주로 다뤄볼 것이다. 이 내용들은 다른 곳에도 매우 많이 사용되니 눈여겨 볼 필요가 있다.

Definition (Step Functions)

A function \(f:[a,b]\rightarrow \mathbb{R}\) is called a step function if it is piecewise constant, i.e., if we have a partition \(P=(x_0,x_1,\ldots,x_n)\) of \([a,b]\) such that
\[ a=x_0<x_1<\cdots <x_n=b. \] and \(f\) is constant on each half open interval \([x_{i-1}, x_i)\) with \(1\le i\le n\). For the step function \(f\), we define the integral to be \[ \int_a^b f(x)dx =\sum_{i=1}^n f(x_i)(x_i-x_{i-1}), \]

Lemma

If \(f\) and \(g\) are step functions on the interval \([a,b]\) with \(f(x)\le g(x)\) for all \(x\in [a,b]\), then \[ \int_a^b f(x)dx \le \int_a^b g(x)dx. \]

• A sketch of proof: let \(A_i=[x_{i-1},x_i)\). Because \(f(x)\le g(x)\) for all \(x\in[a,b]\), \[ \int_{A_i}f(x)dx \le \int_{A_i}g(x)dx. \] This implies the lemma.

Definition (Upper and Lower Integrals)

The lower integral of a function \(f\) on the interval \([a,b]\) is defined by \[ L(f,a,b)=\sup\Big\{\int_a^b s(x)dx : s \mbox{ is a step function with }s\le f \Big\}. \] Similarly, the upper integral of \(f\) is defined to be \[ U(f,a,b)=\inf\Big\{ \int_a^b t(x)dx : t \mbox{ is a step function with } t\ge f\Big\}. \]

• 즉, Greatest Lower Bound, Least Upper Bound의 개념이다.

Lemma

For any bounded function \(f:[a,b]\rightarrow \mathbb{R}\), we have \(L(f,a,b)\le U(f,a,b)\).

• proof : Define \(t(x)\) as a step function on \([a,b]\) with \(t(x)\ge f(x)\) for all \(x\in[a,b]\). If \(s\) is any step function with \(s(x)\le f(x)\) for all \(x\in[a,b]\), then \(s(x)\le t(x)\) for all \(x\in[a,b]\), and hence by the lemma above, \[ \int_a^b s(x)dx\le \int_a^b t(x)dx. \] This implies that the integral of \(t(x)\) is an upper bound for the integrals of any \(s\le f\). Hence \[ ∫^b_a t(x)dx \ge \sup\Big\{\int^b_a s(x): s \mbox{ is a step function with } s\le f\Big\}=L(f,a,b). \] Thus, the integral of any step function \(t\) with \(t\ge f\) is bounded from below by \(L(f,a,b)\). It follows that the greatest lower bound for \(∫^b_a t(x)dx\) with \(t\ge f\) satisfies \(L(f,a,b) \le\inf\Big\{\int^b_a t(x)dx :t \mbox{ is a step function with } t\ge f\Big\}=U(f,a,b)\).

Definition

The function \(f\) is said to be Riemann integrable if its lower and upper integral are the same, i.e., \[ \int^b_a f(x)dx=L(f,a,b)=U(f,a,b) \]

• If we define \(A_i=[x_i-x_{i-1})\) and \(\delta=\sup_{1\le i\le n}A_i\) where \(P=(x_0,x_1\ldots,x_n)\) is a partition of \([a,b]\), then the Riemann integrability is meant to say, similarly, \[ \liminf_{\delta\rightarrow 0\\ n\rightarrow \infty} \sum_{i=1}^n f(x_i)=\lim_{\delta\rightarrow 0\\ n\rightarrow \infty} \sum_{i=1}^n f(x_i)=\limsup_{\delta\rightarrow 0\\ n\rightarrow \infty} \sum_{i=1}^n f(x_i). \]
  

  • 즉, 바꿔 말하면 파티션을 무한히 잘게 쪼개었을 때, integral값의 liminf와 limsup이 하나로 수렴할 때 우리는 Riemann integrable하다고 말할 수 있다.

Proposition

A function \(f:[a,b]→\mathbb{R}\) is Riemann integrable if for every \(\epsilon>0\) there exist step functions \(s,t:[a,b]→\mathbb{R}\) such that for all \(x\in[a,b]\), \(s(x)\le f(x)\le t(x)\) and \[ ∫^b_at(x)dx−∫^b_a s(x)dx<ϵ. \]

Theorem (The Fundamental Theorem of Calculus)

Let \(f:[a,b]\rightarrow \mathbb{R}\) be a continuous and real-valued function. Then, \[ \frac{d}{dx}\int^x_a f (s)ds = f(x). \]

• Proof: By the definition of differentiation, we have \[ \begin{align*} \frac{d}{dx}\int^x_a f (s)ds &= \lim_{h \rightarrow 0} \frac{ \int_a^{x+h}f (s)ds −\int_a^{x}f (s)ds}{h}\\ &= \lim_{h \rightarrow 0} \frac{ \int_x^{x+h}f (s)ds}{h}\\ &= \lim_{h \rightarrow 0} \frac{ hf (x)}{h}\\ &= f(x) \mbox{ }\mbox{ }\mbox{ by the L'Hôpital's rule.} \end{align*} \]

Corollary

Let \(f:[a,b]\rightarrow \mathbb{R}\) be a continuous and real-valued function, and let \(F\) denote an anti-derivative of \(f\) in \([a,b]\). Then, \[ \int_{a}^{b}f(t) dt=F(b)-F(a). \]

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