The primitive intuition of a continuous process is that of one in which small
changes in the input produce small, 'non-catastrophic' changes in the
corresponding output. This idea formalizes easily and naturally for mappings
from one metric space to another: f is continuous at a point p in such
a setting whenever we can force the distance between f(x) and f(p) to be as
small as is desired, merely by taking the distance between x and p to be
small enough. That form of definition is useless in the absence of a properly
defined 'distance' function but, fortunately, it is equivalent to the demand
that the preimage of each open subset of the target metric space shall be open
in the domain. Thus expressed, the idea is immediately transferrable to general
topology:
Examples
(i) If 
is discrete and 
is an arbitrary
topological space, then any function 
is continuous!
Again, if 
is an arbitrary topological space and
is trivial, any mapping
is continuous.
(ii) If , are arbitrary topological spaces
and
is a constant map, then f is continuous.
(iii) Let X be an arbitrary set having more than two elements, with 
. Let 
,
in the definition of continuity; then the identity
map 
is
continuous. However, if we interchange 
with 
so that 
and 
, then
is not continuous! Note that 
is continuous if and only if 
is finer than 
.
Proof Immediate.
There are several different ways to 'recognise' continuity for a mapping
between topological spaces, of which the next theorem indicates two of the
most useful apart from the definition itself:
Proof It is easy to see that (i) implies (ii). Assuming that (ii)
holds, apply it to the closed set 
and (iii) readily follows. Now if
(iii) is assumed and G is a given open subset of 
, use (iii) on the set
and verify that it follows that 
must be open.