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Mistura Creasoti is a creosote-containing mixture. The dispersed or dissolved preparation differs from undiluted creosote material.^[ref]

Copaiba oil is the volatile-oil fraction associated with copaiba oleoresin. Separation of the volatile fraction changes the composition relative to the crude material.

Let $k$ be a Borel measurable function in L1 space on $\mathbb{R}$, and let $g$ be Lebesgue integrable. The function $f(x,y)=k(x-y)g(y)$ is measurable. Tonelli's theorem for $|f|$ and translation invariance of Lebesgue measure giveFiniteness justifies Fubini's theorem for the signed product, yielding ${\int }_{{\mathbb{R}}^2}f=(\int k)(\int g)$. In particular an integrable odd function $k$ makes this signed total zero while the joint L1 norm can remain positive. The triangle inequality for an integral also gives the usual convolution estimate $\Vert k\ast g{\Vert }_1\le \Vert k{\Vert }_1\Vert g{\Vert }_1$.

For an external viewpoint in general position, the horizon is the cycle of edges separating visible and nonvisible faces of a convex polyhedron. In the incremental three-dimensional convex hull algorithm, retain these edges, remove visible faces, and join the viewpoint to each horizon edge to form the new faces. Edges incident with two removed faces disappear.

A continuous dense map into a compact Hausdorff space assigns to each ultrafilter on its domain the unique limit of its pushforward. This limit map is surjective by the ultrafilter characterization of image closure. It is injective exactly when disjoint domain subsets have disjoint image closures.

For real random variables on one probability space, fix a continuity point $x$ of the cumulative distribution function $F_X$. For every $\epsilon >0$, inclusions of events giveTake the lower and upper limits as $n\to \infty$, using convergence in probability, and then let $\epsilon \downarrow 0$. Continuity at $x$ makes both bounds $F_X(x)$, proving convergence in distribution. The two modes of convergence have different information: convergence in probability concerns the coupling on one probability space, whereas convergence in distribution concerns only the marginal probability distributions.

Modulo equality almost everywhere, the displayed metric induces convergence in measure and is complete. A sequence Cauchy in measure has a strictly increasing subsequence $f_{n_k}$ with $\mu (|f_{n_{k+1}}-f_{n_k}|>2^{-k})<2^{-k}$. Subadditivity makes the limsup of these exceptional sets null, so the subsequence has summable pointwise successive differences almost everywhere. It therefore has a finite measurable limit. Almost-everywhere convergence implies convergence in measure for finite measure, and the triangle inequality plus the original Cauchy property extends convergence to the full sequence.

For the metric equivalence, $d(f_n,f)\le \epsilon \mu (E)+\mu (|f_n-f|>\epsilon )$ for $0<\epsilon <1$, while $\mu (|f_n-f|>\epsilon )\le d(f_n,f)/\min \{1,\epsilon \}$. The same estimates compare the two Cauchy properties. Define the subsequence limit as zero where no finite limit exists; that exceptional set is measurable by the countable Cauchy criterion, so completeness of the underlying measure is unnecessary.

On the maximum product metric, the displayed map is nonexpansive when $f$ is a contraction mapping. If the space has two distinct points, changing only the second input coordinate shows that one step preserves some nonzero distance exactly, so it is not itself a strict contraction. Two iterations apply $f$ to both coordinates and therefore contract with the same constant as $f$. This provides a direct example of a map whose iterate is contractive although the map is not.

A random variable with a probability density function $f$, so its cumulative distribution function is $F(x)={\int }_{-\infty }^{x}f(t)dt$. Equivalently, its law is an absolutely continuous probability distribution. In particular, the probability of every singleton is zero.

A probability distribution is singular continuous if it has no point masses and assigns probability one to a set of zero Lebesgue measure. Its cumulative distribution function is continuous, but the distribution has no probability density function with respect to Lebesgue measure. This distinguishes continuity of the cumulative distribution function from an absolutely continuous probability distribution, whose probabilities are integrals of a probability density function.

The unit-radius Wigner semicircle distribution has the displayed probability density function. Its graph is a rescaled upper semicircle. The area of the unit semicircle is $\pi /2$, so the factor $2/\pi$ normalizes it. Symmetry gives mean zero, and the substitution $x=\sin \theta$ gives variance $1/4$.

A scaled radius-$R$ version has density $2\sqrt{R^2-x^2}/(\pi R^2)$ on $(-R,R)$ and variance $R^2/4$. Rejection sampling from a uniform proposal on $(-1,1)$ accepts a proposed $x$ with probability $\sqrt{1-x^2}$; its average acceptance probability is $\pi /4$. Thus the number of proposals for one output has geometric distribution with expectation $4/\pi$. This law also describes the limiting empirical eigenvalue density in standard ensembles of large symmetric random matrices.

For a scale $a>0$ and shape $\kappa >0$, the log-logistic distribution has cumulative distribution function and probability density functionIt has no point mass at zero. Its quantile function is $Q(p)=a[p/(1-p){]}^{1/\kappa }$ for $0<p<1$, so its median is $a$. The log odds relation $\log (F(x)/(1-F(x)))=\kappa \log x-\kappa \log a$ explains both its name and the log-logistic probability plot.^[ref]

Its survival function is $1/(1+(x/a{)}^{\kappa })$, with power tail proportional to $x^{-\kappa }$. A positive moment $\mathbb{E}{X}^r$ is finite exactly when $r<\kappa$: near infinity its defining integral behaves like $\int x^{r-\kappa -1}dx$. For example, $a=2,\kappa =2$ gives median $2$ and $\mathbb{P}(X>4)=1/5$. Writing $a=\alpha$ and $\kappa =\beta -1$ is an alternative parameterization requiring $\beta >1$.

A continuous map from a nonempty finite product space of infinite sets with their cofinite topologies to a Hausdorff space is a constant function. Fixing all but one coordinate gives a continuous map from an infinite cofinite topology, so changing that coordinate does not change the value. Any two points in a finite Cartesian product can be joined by finitely many one-coordinate changes. This proves the assertion. In particular every real-valued continuous function on $Z\times Z$ is constant when $Z$ is the real line equipped with the cofinite topology.

Let $Z$ be a nonempty infinite set with the cofinite topology, let $T$ be any topological space, and let $Y$ be Hausdorff. For a continuous map $F:Z\times T\to Y$, every slice $z\mapsto F(z,t)$ is constant. Thus $F(z,t)=g(t)$, where $g(t)=F(z_0,t)$ for any fixed $z_0$. The map $g$ is continuous, since $t\mapsto (z_0,t)$ is continuous into the product topology. It is unique because the coordinate projection ${\pi }_T$ is a surjection. If $F$ is a surjection, so is $g$.

For $T=[0,1]$ and $Y$ homeomorphic to the circle, a surjective $g$ cannot be injective: otherwise the compact-to-Hausdorff continuous bijection theorem would contradict the fact that the circle and interval are not homeomorphic. Hence distinct $a,b\in [0,1]$ satisfy $F(z,a)=F(z,b)$ simultaneously for all $z\in Z$.

A function $f:X\to Y$ between topological spaces is continuous exactly when the inverse image of every closed set in $Y$ is closed in $X$. The defining open set criterion is equivalent, because $f^{-1}(Y\setminus A)=X\setminus f^{-1}(A)$. This criterion concerns inverse images; a continuous map need not be a closed map.

A function $f:X\to Y$ between topological spaces is continuous if and only if $f(\overline{A})\subseteq \overline{f(A)}$ for every subset $A\subseteq X$, where the bars denote closure. For a continuous $f$, pull back an open neighbourhood of $f(x)$ for $x\in \overline{A}$; its preimage meets $A$, so the neighbourhood meets $f(A)$. Conversely, apply the inclusion to $A=f^{-1}(C)$ with $C$ closed. Then $f(\overline{A})\subseteq C$ implies $\overline{A}\subseteq A$, so the preimage of every closed subset is closed and $f$ is continuous.

A function of several variables is separately continuous when it is continuous in each variable with the others fixed. This is weaker than joint continuity in the product topology. Functions with narrow rectangular supports provide useful examples distinguishing convergence in topologies generated by separate continuity from Euclidean convergence.

For a continuous map into a metric space, $f(\overline{A})\subseteq \overline{f(A)}$: an open neighborhood of an image point pulls back to a neighborhood meeting $A$. A bounded set lies in a closed ball, and its closure lies in the same ball because distance from the centre is a continuous function. The displayed boundedness conclusion follows. Neither compactness of the domain nor uniform continuity of the map is needed.

A bounded linear operator between normed spaces is approximable if it is an operator norm limit of bounded finite-rank operators. If the target is a Banach space, every approximable operator is a compact operator. For a given tolerance, approximate the operator in norm by one finite-rank operator, cover that operator's bounded unit-ball image by finitely many small balls in its finite-dimensional range, and transfer the cover across the operator-norm error. This makes the original image totally bounded; its closure is complete in the Banach target and therefore compact. The converse on arbitrary Banach spaces would require an additional approximation hypothesis and is not part of this assertion.

An isomorphic embedding of normed vector spaces is an injective bounded linear operator whose inverse on its image is bounded. Equivalently $\Vert Tx\Vert \ge c\Vert x\Vert$ for some $c>0$. The two-sided bounds make the domain and its image carry equivalent norms. If the domain is a Banach space, a convergent sequence $T{x}_n$ has a Cauchy sequence of preimages, so its limit remains in the image; thus the image is closed.