## 1.5 Basic analytical tools

We will make use of the following well-known analytical results.

Theorem 1.9 (Mean value theorem) Let $$f:[a,b]\longrightarrow\mathbb{R}$$ be a continuous function and differentiable in $$(a,b)$$. Then there exists $$c\in(a,b)$$ such that $$f(b)-f(a)=f'(c)(b-a)$$.
Theorem 1.10 (Integral mean value theorem) Let $$f:[a,b]\longrightarrow\mathbb{R}$$ be a continuous function over $$(a,b)$$. Then there exists $$c\in(a,b)$$ such that $$\int_a^b f(x)\,\mathrm{d}x=f(c)(b-a)$$.

Theorem 1.11 (Taylor’s theorem) Let $$f:\mathbb{R}\longrightarrow\mathbb{R}$$ and $$x\in\mathbb{R}$$. Assume that $$f$$ has $$r$$ continuous derivatives in an interval $$(x-\delta,x+\delta)$$ for some $$\delta>0$$. Then for any $$0<h<\delta$$,

\begin{align*} f(x+h)=\sum_{j=0}^r\frac{1}{j!}f^{(j)}(x)h^j+R_r,\quad R_r=o(h^r). \end{align*}

Remark. The remaider $$R_r$$ depends on $$x\in\mathbb{R}$$. Explicit control of $$R_r$$ is possible if $$f$$ is further assumed to be $$(r+1)$$ differentiable in $$(x-\delta,x+\delta)$$. In this case, $$R_r=\frac{f^{(r+1)}(\xi_x)}{(r+1)!}h^{r+1}=o(h^r)$$, for some $$\xi_x\in(x-\delta,x+\delta)$$. Then, if $$f^{(r+1)}$$ is bounded in $$(x-\delta,x+\delta)$$, $$\sup_{y\in(x-\delta,x+\delta)}\frac{R_r}{h^r}\to0$$, i.e., the remainder is $$o(h^r)$$ uniformly in $$(x-\delta,x+\delta)$$.

Theorem 1.12 (Dominated Convergence Theorem; DCT) Let $$f_n:S\subset\mathbb{R}\longrightarrow\mathbb{R}$$ be a sequence of Lebesgue measurable functions such that $$\lim_{n\to\infty}f_n(x)=f(x)$$ and $$|f_n(x)|\leq g(x)$$, $$\forall x\in S$$ and $$\forall n\in\mathbb{N}$$, where $$\int_S |g(x)|\,\mathrm{d}x<\infty$$. Then

\begin{align*} \lim_{n\to\infty}\int_S f_n(x)\,\mathrm{d}x=\int_S f(x)\,\mathrm{d}x<\infty. \end{align*}

Remark. Note that if $$S$$ is bounded and $$|f_n(x)|\leq M$$, $$\forall x\in S$$ and $$\forall n\in\mathbb{N}$$, then limit interchangeability with integral is always possible.