In real and functional analysis, equicontinuity is a concept which extends the notion of uniform continuity from a single function to collection of functions.
Given topological vector spaces 
and 
, a collection 
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 
. In the special case that 
is a metric space and
, this criterion can be restated as
an epsilon-delta definition: A collection
of real-valued continuous functions on 
is equicontinuous if, given 
, there is a 
such that whenever 
satisfy 
,
 |
for all 
.
It is often convenient to visualize an equicontinuous collection of functions as
being "uniformly uniformly continuous," i.e., a collection 
for which a single 
can be chosen for any arbitrary 
so as to make all 
uniformly continuous simultaneously, independent
of 
.
In the latter case, equicontinuity is the ingredient needed to "upgrade" pointwise convergence to uniform
convergence, i.e., an equicontinuous sequence 
of functions which converges pointwise
to a function 
actually converges uniformly to 
.
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
(rather than linear) maps 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 
.
