Infi-2 5

Linearity of definite integral #lemma

\begin{matrix} \int_{a}^{b} (α f + g) (x) d x = α \int_{a}^{b} f (x) d x + \int_{a}^{b} g (x) d x \\ Proof: \\ \sum (α f + g) (c_{i}) Δ x_{i} = α \sum f (c_{i}) Δ x_{i} + \sum g (c_{i}) Δ x_{i} \end{matrix}

Monotonicity of definite integral #lemma

\begin{matrix} f (x) \leq g (x) ⟹ \int_{a}^{b} f (x) d x \leq \int_{a}^{b} g (x) d x \\ 0 \leq f (x) ⟹ 0 \leq \int_{a}^{b} f (x) d x \\ m \leq f (x) \leq M ⟹ m (b - a) \leq \int_{a}^{b} f (x) d x \leq M (b - a) \end{matrix}

Summation of definite integrals #lemma

\begin{matrix} \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x = \int_{a}^{c} f (x) d x \end{matrix}

Absolute value of definite integral #lemma

\begin{matrix} | \int_{a}^{b} f (x) d x | \leq \int_{a}^{b} | f (x) | d x \end{matrix}

\begin{matrix} \int_{0}^{2 π} \sin (x) d x = 0 \\ \int_{0}^{2 π} | \sin (x) | d x = 4 \end{matrix}

Continuous function is Riemann-integrable #theorem

\begin{matrix} Let f be a continuous function on [a, b] \\ Then f is Riemann-integrable \\ Proof: \\ Let {a_{n}}, {b_{n}} \subseteq [a, b] : a_{n} - b_{n} \to 0 \\ f is continuous ⟹ f (a_{n}) - f (b_{n}) \to 0 \\ Let {P_{n}} : λ (P_{n}) \to 0 \\ Let ω_{k} be a maximal oscilate in P_{n} \\ ⟹ \sum ω_{i} Δ x_{i} \leq \sum ω_{k} Δ x_{i} = ω_{k} \sum Δ x_{i} = ω_{k} (b - a) \\ 0 \leq \sum ω_{i} Δ x_{i} \leq ω_{k} (b - a) \\ ω_{k} = sup_{[x_{k - 1}, x_{k}]} f - inf_{[x_{k - 1}, x_{k}]} f \\ By Weierstrass theorem: {\begin{matrix} sup_{[x_{k - 1}, x_{k}]} f = max_{[x_{k - 1}, x_{k}]} f \\ inf_{[x_{k - 1}, x_{k}]} f = min_{[x_{k - 1}, x_{k}]} f \end{matrix} \\ ⟹ \exists m_{k}, M_{k} \in [x_{k - 1}, x_{k}] : ω_{k} = sup_{[x_{k - 1}, x_{k}]} f - inf_{[x_{k - 1}, x_{k}]} f = f (M_{k}) - f (m_{k}) \\ λ (P_{n}) \to 0 \\ 0 \leq x_{k} - x_{k - 1} \leq λ (P_{n}) ⟹ x_{k} - x_{k - 1} \to 0 \\ 0 \leq | M_{k} - m_{k} | \leq x_{k} - x_{k - 1} ⟹ | M_{k} - m_{k} | \to 0 ⟹ M_{k} - m_{k} \to 0 \\ ⟹ f (M_{k}) - f (m_{k}) \to 0 ⟹ ω_{k} \to 0 \\ ⟹ ω_{k} (b - a) \to 0 ⟹ \sum ω_{i} Δ x_{i} \to 0 \\ ⟹ By Lebesgue criterion: f is Riemann-integrable \end{matrix}

Bounded function with a finite number of discontinuities is Riemann-integrable #theorem

\begin{matrix} f is Riemann-integrable ⟺ f is bounded and has a finite number of discontinuities \\ Explanation (not proof): \\ ⟹ Let f be bounded \\ Let there be one discontinuity C \\ Let {P_{n}} : λ (P_{n}) \to 0 \\ Let C \in [x_{k - 1}, x_{k}] \\ \sum ω_{i} Δ x_{i} = \sum_{i < k} ω_{i} Δ x_{i} + ω_{k} Δ x_{k} + \sum_{i > k} ω_{i} Δ x_{i} \\ f is bounded ⟹ ω_{k} is finite \\ Δ x_{k} \to 0 ⟹ ω_{k} Δ x_{k} \to 0 \\ \sum_{i < k} ω_{i} Δ x_{i} \to 0 (see previous theorem) \\ \sum_{i > k} ω_{i} Δ x_{i} \to 0 (see previous theorem) \\ ⟹ \sum ω_{i} Δ x_{i} \to 0 \\ If number of discontinuities is finite, there is a finite number of such ω_{k_{j}} Δ x_{k_{j}} \\ that all tend to 0 ⟹ Their sum also tends to 0 \\ ⟹ f is Riemann-integrable \\ ⟸ Let f be Riemann-integrable \\ Let {P_{n}} : λ (P_{n}) \to 0 \\ ⟹ \sum ω_{i} Δ x_{i} \to 0 \\ Let D = {k} \subseteq [1, n] be a set of intervals with discontinuities \\ \sum ω_{i} Δ x_{i} = \sum_{i \notin D} ω_{i} Δ x_{i} + \sum_{k \in D} ω_{k} Δ x_{k} \\ \sum_{i \notin D} ω_{i} Δ x_{i} \to 0 ⟹ \sum_{k \in D} ω_{k} Δ x_{k} \to 0 \\ And this is only possible when number of discontinuitites is finite \end{matrix}