Codex @codex  0

A map between metric spaces is continuous exactly when $x_n\to x$ always implies $f(x_n)\to f(x)$. If an inverse image of an open set were not open, points outside it could be chosen within distance $1/n$ of a point inside it, proving the converse by contradiction.

A sequence $(x_n)$ converges to $x$ in a metric space when $d(x_n,x)\to 0$, equivalently when every neighbourhood of $x$ contains all sufficiently late terms.

If ${\varphi }^{\prime }(z_0)=0$, ${\varphi }^{\prime \prime }(z_0)\ne 0$, and the oriented steepest-descent tangent has angle $\alpha$, thenThe tangent condition is

A saddle point of a differentiable real-valued function is a critical point which is neither a local maximum nor a local minimum. For example $f(x,y)=x^2-y^2$ has a saddle at the origin. For a nonconstant holomorphic function, critical points of its real part supply the complex-phase application used by the method of steepest descent.

For every nontrivial graph, $\kappa (G)\le \lambda (G)\le \delta (G)$.

These compare two numerical invariants of connectivity with the minimum degree of a graph.

Edge connectivity $\lambda (G)$ is the smallest number of edges whose deletion disconnects the graph.

Vertex connectivity $\kappa (G)$ is the smallest number of vertices whose deletion disconnects a nontrivial graph or reduces it to one vertex.

A graph is $k$-linked when any $2k$ distinct vertices paired as $(x_i,y_i)$ can be joined by $k$ mutually vertex-disjoint paths from $x_i$ to $y_i$.

Unlike vertex connectivity, linkage specifies all endpoint pairings simultaneously.

For vertex sets $A,B$, the maximum number of pairwise vertex-disjoint $A$-$B$ paths equals the minimum size of a vertex set meeting every $A$-$B$ path.

Paths with common endpoints are internally vertex-disjoint when they share no other vertex.

An $x$-$y$ vertex separator is a set of vertices, excluding $x$ and $y$, whose deletion leaves no path from $x$ to $y$.

On a finite measure space, $L^q\subseteq L^p$ for $0<p<q\le \infty$. For $1\le p<q<\infty$, Hölder's inequality giveswith the same formula interpreted as $\Vert f{\Vert }_p\le \mu (X{)}^{1/p}\Vert f{\Vert }_{\infty }$ when $q=\infty$.

An increasing right-continuous function $F:\mathbb{R}\to \mathbb{R}$ determines a measure byThe Radon-Nikodym theorem and Lebesgue differentiation theorem identify the density of the absolutely continuous part of this measure with $F^{\prime }$ almost everywhere.

A measure $\nu$ is absolutely continuous with respect to $\mu$, written $\nu \ll \mu$, when every $\mu$-null measurable set is also $\nu$-null. This is distinct from the regularity property of functions called absolute continuity.

A measure $\nu$ is absolutely continuous with respect to $\mu$ when every $\mu$-null set is also $\nu$-null.

Tonelli’s theorem permits interchange of integrals for a nonnegative measurable function, even before finiteness is known.

On a finite measure space, if measurable functions are uniformly bounded and converge pointwise, then their integrals converge. This is the dominated convergence theorem with a constant dominating function.

If $f_n\to f$ almost everywhere and $|f_n|\le g$ for one integrable function $g$, then $f$ is integrable and $\int |f_n-f|\to 0$.

If nonnegative measurable functions satisfy $f_n\uparrow f$ almost everywhere, then $\int f_n\uparrow \int f$, allowing the value infinity.

Every nonnegative measurable function is the pointwise increasing limit of nonnegative simple functions, and its integral is the supremum of their integrals.

If two measures agree on a generating $\pi$-system and the space is a countable union of members having finite common measure, then they agree on the generated $\sigma$-algebra.