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.