1.5 Review of 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 $p$ continuous derivatives in an interval $(x-\delta,x+\delta)$ for a $\delta>0.$ Then, for any $|h|<\delta,$

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

Remark. The remainder $R_n$ depends on $x\in\mathbb{R}.$ Explicit control of $R_n$ is possible if $f$ is further assumed to be $(p+1)$ differentiable in $(x-\delta,x+\delta).$ In that case, $R_n=\frac{f^{(p+1)}(\xi_x)}{(p+1)!}h^{p+1}=o(h^p)$ for a certain $\xi_x\in(x-\delta,x+\delta).$ Then, if $f^{(p+1)}$ is bounded in $(x-\delta,x+\delta),$ $\sup_{y\in(x-\delta,x+\delta)}\frac{R_n}{h^p}\to0,$ i.e., the remainder is $o(h^p)$ uniformly in $(x-\delta,x+\delta).$ The remainder can also be put in an integral form that provides further control of the error term.

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.