Consider a sequence
of continuous real-valued functions of a real variable. The sequence
converges pointwise on a set
to a function
if for each
in
,
as
. The limit
is not guaranteed to be continuous; in this Demonstration the limit has a removable discontinuity. (To construct a limit that is discontinuous everywhere in
, construct
with spikes at all numbers that can be written in the form
, where
and
are positive integers and
.)
The limit of a
uniformly convergent sequence of continuous functions is guaranteed to be continuous. Here "uniformly" means that in the ϵ-
definition of the limit, the same
must apply to
every 
in
. Pointwise convergence only requires an
that may depend on
.
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