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A metric space is complete when every Cauchy sequence converges to a point of the space.

A metric space $X$ is compact exactly when every continuous function $X\to \mathbb{R}$ is bounded. For the converse, a noncompact metric space contains a closed discrete sequence; the unbounded function assigning its $n$th point the value $n$ extends to $X$ by the Tietze extension theorem.

Every metric space is normal. For disjoint closed sets $A,B$, the functionis continuous, equals zero on $A$, and equals one on $B$; inverse images of disjoint neighbourhoods of zero and one separate the sets.

Every sequence in a compact metric space has a convergent subsequence. Conversely, sequential compactness and compactness are equivalent for metric spaces.

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.