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continuous relation
The idea of a continuous relation is neither as old nor as wellestablished as the idea of a continuous function. Different authors use somewhat different definitions. The present article is based on the following definition:
Let $X$ and $Y$ be topological spaces and $R$ a relation between $X$ and $Y$ ($R$ is a subset of $X\times Y$). $R$ is said to be continuous if
for any open subset $V$ of $Y$, $R^{{1}}(V)$ is open in $X$.
Here $R^{{1}}(V)$ is the inverse image of $V$ under $R$, and is defined as
$R^{{1}}(V):=\{x\in X\mid xRy\mbox{ for some }y\in V\}.$ 
Equivalently, $R$ is a continuous relation if for any open set $V$ of $Y$, the set $\pi_{X}((X\times V)\cap R)$ is open in $X$, where $\pi_{X}$ is the projection map $X\times Y\to X$.
Remark. Continuous relations are generalization of continuous functions: if a continuous relation is also a function, then it is a continuous function.
Some examples.

Let $X$ be an ordered space. Then the partial order $\leq$ is continuous iff for every open subset $A$ of $X$, its lower set $\downarrow\!\!A$ is also open in $X$.
In particular, in $\mathbb{R}$, the usual linear ordering $\leq$ on $\mathbb{R}$ is continuous. To see this, let $A$ be an open subset of $\mathbb{R}$. If $A=\varnothing$, then $\downarrow\!\!A=\varnothing$ as well, and so is open. Suppose now that $A$ is nonempty and deal with the case when $A$ is not bounded from above. If $r\in\mathbb{R}$, then there is $a\in A$ such that $r\leq a$, so that $r\in\downarrow\!\!A$, which implies $\downarrow\!\!A=\mathbb{R}$. Hence $\downarrow\!\!A$ is open. If $A$ is bounded from above, then $A$ has a supremum (since $\mathbb{R}$ is Dedekind complete), say $x$. Since $A$ is open, $x\notin A$ (or else $x\in(a,b)\subseteq A$, implying $x<\frac{x+b}{2}\in(a,b)$, contradicting the fact that $x$ is the least upper bound of $A$). So $\downarrow\!\!A=(\infty,x)$, which is open also. Therefore, $\leq$ is a continuous relation on $\mathbb{R}$.

Again, we look at the space $\mathbb{R}$ with its usual interval topology. The relation this time is $R=\{(x,y)\in\mathbb{R}^{2}\mid x^{2}+y^{2}=1\}$. This is not a continuous relation. Take $A=(2,2)$, which is open. But then $R^{{1}}(A)=[1,1]$, which is closed.

Now, let $X$ be a locally connected topological space. For any $x,y\in X$, define $x\sim y$ iff $x$ and $y$ belong to the same connected component of $X$. Let $A$ be an open subset of $X$. Then $B=\sim^{{1}}(A)$ is the union of all connected components containing points of $A$. Since (it can be shown) each connected component is open, so is their union, and hence $B$ is open. Thus $\sim$ is a continuous relation.

If $R$ is symmetric, then $R$ is continuous iff $R^{{1}}$ is. In particular, in a topological space $X$, an equivalence relation $\sim$ on $X$ is continuous iff the projection $p$ of $X$ onto the quotient space $X/\sim$ is an open mapping.
Remark. Alternative definitions: One apparently common definition (as described by Wyler) is to require inverse images of open sets to be open and inverse images of closed sets to be closed (making the relation upper and lower semicontinuous). Wyler suggests the following definition: If $r\colon e\to f$ is a relation between topological spaces $E$ and $F$, then $r$ is continuous iff for each topological space $A$, and functions $f\colon A\to E$ and $g\colon A\to F$ such that $f(u)\mathrel{r}g(u)$ for all $u\in A$, continuity of $f$ implies continuity of $g$.
References
 1 T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, New York (2005).
 2 J. L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
 3 Oswald Wyler, A Characterization of Regularity in Topology Proceedings of the American Mathematical Society, Vol. 29, No. 3. (Aug., 1971), pp. 588590.
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Comments
Varying definitions
There seem to be a several definitions of "continuous relation" flying around. Inverse images of closed sets may be required to be closed, and images of points may even be required to be compact. Is there any definition that is particularly popular? Has there been much work on this topic? To what extent do theorems about continuous functions extend to continuous relations (with various definitions)?
Re: Varying definitions
Well, there is huge theory about multimaps (short for multivalued maps). Multimap F:XoY between topological spaces X,Y is nothing else then function F:X>2^Y (some authors require some additonal properties, for example F(x) to be nonempty and compact for all x\in X, but this is not necessary). If you have relation R (subset of XxY), then you can look at it as a multimap via this onetoone correspondence (between relations on XxY and multimaps from subspaces of X to Y):
define X_R={x\inX; \exists y such that xRy}, then define F_R:X_RoY as follows:
F_R(x):={y\inY; xRy}.
Of course essentialy there's no difference between multimaps and relations, but sometimes it is easier to think about relation as a function.
As I said there is huge theory about multimaps. There you can talk about upper semicontinous multimaps, lower semicontinous multimaps, etc. These are good properties, but property of relation to be continous (in the sense you were refering to) is not good (in my opinion), because it is too strong.
joking
Re: Varying definitions
> but property of relation to be continous (in the sense you were
> refering to) is not good (in my opinion), because it is too strong.
Ups... this property is exactly lower semicontinuity for multimaps. :) I was thinking about something else.
joking
Re: Varying definitions
For the final definition of continuousness (applying to any morphisms of certain categories) see my article "Generalized Continuousness" at
http://www.mathematics21.org/binaries/continuousness.pdf
Actually, to completely understand that article you need reading some other prerequisites from Algebraic General Topology
http://www.mathematics21.org/algebraicgeneraltopology.html
For multivalued functions (well, multivalued morphisms in general) there are three distinct definitions of continuousness.
P.S. Nominate me for Abel Prize  http://www.mathematics21.org/abelprize.html

Victor Porton  http://www.mathematics21.org
* Algebraic General Topology and Math Synthesis
* 21 Century Math Method (post axiomatic math logic)
* Category Theory  new concepts
Oops.. I broke it.
It looks like I introduced an error in the LaTeX, but I can't find it!
Re: Oops.. I broke it.
And removed the error. Unfortunately along with the bibliography entry I was trying to add. Let's see if I can make this happen.
Re: Oops.. I broke it.
All better. Now all that's left is adding proper links and a proper footnote or whatever to reference my source.