B Preliminaries from calculus

B.1 Differentiability

Definition B.1 Real differentiability

Let \(m,n \in \mathbb {N}\), and let \(f \colon U \to \mathbb {R}^m\) be a function defined on a subset \(U \subset \mathbb {R}^n\). A linear map \(L \colon \mathbb {R}^n \to \mathbb {R}^m\) is said to be a differential of \(f\) at \(p_0 \in U\) if

\begin{align*} f(p) = f(p_0) + L (p-p_0) + E(p-p_0) \end{align*}

where the error term \(E\) is small near \(p_0\) in the sense that

\begin{align*} \lim _{p \to p_0} \frac{\| E(p-p_0) \| }{\| p-p_0\| } = 0 . \end{align*}

We say that \(f\) is differentiable at \(p_0\) if such a linear map \(L\) exists.

It is easy to check that the differential \(L\) of \(f\) at \(p_0\) is unique if \(p_0\) is an interior point of \(U\); we then denote it by \(L = \mathrm{d}f (p_0)\).

Lemma B.2 Differentiability implies continuity

If a function \(f \colon U \to \mathbb {R}^m\) defined on a subset \(U \subset \mathbb {R}^n\) is differentiable at \(p_0 \in U\), then it is continuous at \(p_0\).

Proof ▶

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Lemma B.3 Jacobian matrix of the differential

If a function \(f \colon U \to \mathbb {R}^m\) defined on a subset \(U \subset \mathbb {R}^n\) is differentiable at an interior point \(p_0\) of \(U\), then it has all first order partial derivatives at \(p_0\), and the matrix representation of the differential \(\mathrm{d}f (p_0)\) in the standard bases of \(\mathbb {R}^m\) and \(\mathbb {R}^n\) is

\begin{align*} \mathrm{d}f (p_0) = \left[ \begin{array}{ccc} \frac{\partial f_1}{\partial x_1}(p_0) & \cdots & \frac{\partial f_1}{\partial x_n}(p_0) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(p_0) & \cdots & \frac{\partial f_m}{\partial x_n}(p_0) \end{array} \right] \in \mathbb {R}^{m \times n} , \end{align*}

where \(f_1, \ldots , f_m \colon U \to \mathbb {R}\) denote component functions of \(f\).

Proof ▶

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Lemma B.4 Vanishing partial derivatives implies locally constant

Suppose that \(f \colon U \to \mathbb {R}^m\) is a function defined on an open and connected subset \(U \subset \mathbb {R}^n\) of \(\mathbb {R}^n\) whose first order partial derivatives exist and are zero at all points of \(U\). Then \(f\) is a constant function.

Proof ▶

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B.2 Riemann integral

For the purposes of this course, it suffices to know the Riemann integral. (Those who already know Lebesgue integration theory can substitute that more general notion of integral everywhere.)

Definition B.5 Riemann integral

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Lemma B.6 Riemann integrability of continuous functions

Any continuous function \(f \colon [a,b] \to \mathbb {R}\) is Riemann integrable on \([a,b]\).

Proof ▶

See MS-C1541 Metric Spaces.

B.3 Trigonometry

Lemma B.7 Trigonometric angle sum identities

Let \(\alpha , \beta \in \mathbb {R}\). Then we have

\begin{align*} \cos (\alpha + \beta ) \; = \; \cos (\alpha ) \, \cos (\beta ) - \sin (\alpha ) \, \sin (\beta ) \\ \sin (\alpha + \beta ) \; = \; \cos (\alpha ) \, \sin (\beta ) + \sin (\alpha ) \, \cos (\beta ) . \end{align*}

Proof ▶

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B.4 Supremum, infimum, limit superior, and limit inferior

Definition B.8 Supremum

The supremum, or the least upper bound, of a set \(A \subset \mathbb {R}\) is the smallest real number \(s\) such that \(a \le s\) for all \(a \in A\), and is denoted by \(s = \sup A\).

By the completeness axiom of real numbers, every nonempty set (\(A \ne \emptyset \)) of real numbers which is bounded from above (for some \(u \in \mathbb {R}\) we have \(a \le u\) for all \(a \in A\)) has a supremum \(\sup A \in \mathbb {R}\). We adopt the notational conventions that \(\sup \emptyset = -\infty \), and that \(\sup A = +\infty \) if \(A\) is not bounded from above.

For convenience, we also adopt some flexibility in the notation: for example the supremum of values of a real-valued function on a set \(D\) is denoted by

\begin{align*} \sup _{x \in D} f(x) \; := \; \sup \big\{ f(x) \; \big| \; x \in D \big\} \end{align*}

and the supremum of values in the tail of a real-number sequence \((x_n)\) starting from index \(m\) is denoted by

\begin{align*} \sup _{n \ge m} x_n \; := \; \sup \big\{ x_n \; \big| \; m \ge n \big\} \, . \end{align*}

Definition B.9 Infimum

The infimum, or the greatest lower bound, of a set \(A \subset \mathbb {R}\) is the greatest real number \(i\) such that \(a \ge i\) for all \(a \in A\), and is denoted by \(i = \inf A\).

By the completeness axiom of real numbers, every nonempty set (\(A \ne \emptyset \)) of real numbers which is bounded from below (for some \(\ell \in \mathbb {R}\) we have \(a \ge \ell \) for all \(a \in A\)) has an infimum \(\inf A \in \mathbb {R}\). We adopt the notational conventions that \(\inf \emptyset = +\infty \), and that \(\inf A = -\infty \) if \(A\) is not bounded from below.

For convenience, we also adopt some flexibility in the notation for infimums of function values or sequence values, similarly as with supremums.

Definition B.10 Limit superior

Let \((x_n)_{n \in \mathbb {N}}\) be a sequence of real numbers. Then the limit superior of the sequence is defined as

\begin{align*} \limsup _{n \to \infty } x_n \; := \; \lim _{m \to \infty } \Big( \sup _{n \ge m} x_n \Big) . \end{align*}

With the following conventions, the limit superior of a sequence always exists as either a real number or one of the symbols \(\pm \infty \). If the sequence is not bounded from above, then by conventions regarding the supremum, we have \(\sup _{n \ge m} x_n = +\infty \) for every \(m\), so we correspondingly set \(\limsup _{n \to \infty } x_n = +\infty \). Otherwise the sequence \((\sup _{n \ge m} x_n)_{m \in \mathbb {N}}\) is a decreasing sequence of real numbers, so either it is bounded from below and converges to \(\lim _{m \to \infty } \big( \sup _{n \ge m} x_n \big) \, = \, \inf _{m \in \mathbb {N}} \big( \sup _{n \ge m} x_n \big) \in \mathbb {R}\), or it is not bounded from below and we set \(\limsup _{n \to \infty } x_n \, = \, \inf _{m \in \mathbb {N}} \big( \sup _{n \ge m} x_n \big) = -\infty \).

Definition B.11 Limit inferior

Let \((x_n)_{n \in \mathbb {N}}\) be a sequence of real numbers. Then the limit inferior of the sequence is defined as

\begin{align*} \liminf _{n \to \infty } x_n \; := \; \lim _{m \to \infty } \Big( \inf _{n \ge m} x_n \Big) . \end{align*}

With the following conventions, the limit inferior of a sequence always exists as either a real number or one of the symbols \(\pm \infty \). If the sequence is not bounded from below, then by conventions regarding the infimum, we have \(\inf _{n \ge m} x_n = -\infty \) for every \(m\), so we correspondingly set \(\liminf _{n \to \infty } x_n = -\infty \). Otherwise the sequence \((\inf _{n \ge m} x_n)_{m \in \mathbb {N}}\) is an increasing sequence of real numbers, so either it is bounded from above and converges to \(\lim _{m \to \infty } \big( \inf _{n \ge m} x_n \big) \, = \, \sup _{m \in \mathbb {N}} \big( \inf _{n \ge m} x_n \big) \in \mathbb {R}\), or it is not bounded from above and we set \(\liminf _{n \to \infty } x_n \, = \, \sup _{m \in \mathbb {N}} \big( \inf _{n \ge m} x_n \big) = +\infty \).

Lemma B.12 Limit with limsup and liminf

Let \((x_n)_{n \in \mathbb {N}}\) be a sequence of real numbers, and let \(x \in \mathbb {R}\). Then the following are equivalent:

The limit \(\lim _{n \to \infty } x_n\) exists and equals \(x\).
We have both \(\limsup _{n \to \infty } x_n = x\) and \(\liminf _{n \to \infty } x_n = x\).

(With the usual conventions of \(\pm \infty \) as possible limits of real-number sequences, the above equivalence of conditions also extends to the cases \(x = \pm \infty \).)

Proof ▶

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