Equicontinuous families of functions share a local continuity condition uniformly over the family. Uniform equicontinuity extends uniform continuity from one function to a family.
Given topological vector spaces 
and 
, a family 
of linear
transformations from 
into 
is said to be equicontinuous if to every neighborhood 
of 
in 
there corresponds a neighborhood 
of 
in 
such that 
for all 
. For a general family 
of real-valued continuous functions on a metric
space 
, equicontinuity at a point 
means that every 
has a 
, possibly depending
on 
, such that 
for every 
whenever 
. The family
is equicontinuous if this holds at every point. The stronger
notion of uniform equicontinuity requires a single 
independent of the point: given
, there is a 
such that whenever 
satisfy 
,
 |
for all 
. It is often convenient
to visualize a uniformly equicontinuous family of functions
as being "uniformly uniformly continuous,"
i.e., a family 
for which a single 
can be chosen for any arbitrary 
so as to make all 
uniformly continuous
simultaneously, independent of 
.
On a compact metric space, equicontinuity implies uniform equicontinuity. On a compact
space 
, an equicontinuous sequence 
of real-valued continuous functions that has pointwise
convergence to 
has uniform convergence to 
. Compactness is essential for this implication from
pointwise convergence to uniform
convergence.
These definitions may be restated to accommodate subtle changes in construction. For example, in the special case that 
is locally convex, 
is a nonempty subset which is compact
and convex, and 
is a group (rather than a set) of affine transformations
(rather than linear transformations) from
into 
, the above definition is modified and 
is said to be equicontinuous if every neighborhood 
of 
in 
corresponds to a neighborhood 
of 
in 
such that 
whenever 
, 
, and 
.
