Uniform approximation #
In this file, we give lemmas ensuring that a function is continuous if it can be approximated
uniformly by continuous functions. We give various versions, within a set or the whole space, at
a single point or at all points, with locally uniform approximation or uniform approximation. All
the statements are derived from a statement about locally uniform approximation within a set at
a point, called continuousWithinAt_of_locally_uniform_approx_of_continuousWithinAt
.
Implementation notes #
Most results hold under weaker assumptions of locally uniform approximation. In a first section, we prove the results under these weaker assumptions. Then, we derive the results on uniform convergence from them.
Tags #
Uniform limit, uniform convergence, tends uniformly to
A function which can be locally uniformly approximated by functions which are continuous within a set at a point is continuous within this set at this point.
A function which can be locally uniformly approximated by functions which are continuous at a point is continuous at this point.
A function which can be locally uniformly approximated by functions which are continuous on a set is continuous on this set.
A function which can be uniformly approximated by functions which are continuous on a set is continuous on this set.
A function which can be locally uniformly approximated by continuous functions is continuous.
A function which can be uniformly approximated by continuous functions is continuous.
Uniform limits #
From the previous statements on uniform approximation, we deduce continuity results for uniform limits.
A locally uniform limit on a set of functions which are continuous on this set is itself continuous on this set.
A uniform limit on a set of functions which are continuous on this set is itself continuous on this set.
A locally uniform limit of continuous functions is continuous.
A uniform limit of continuous functions is continuous.
Composing limits under uniform convergence #
In general, if Fₙ
converges pointwise to a function f
, and gₙ
tends to x
, it is not true
that Fₙ gₙ
tends to f x
. It is true however if the convergence of Fₙ
to f
is uniform. In
this paragraph, we prove variations around this statement.
If Fₙ
converges locally uniformly on a neighborhood of x
within a set s
to a function f
which is continuous at x
within s
, and gₙ
tends to x
within s
, then Fₙ (gₙ)
tends
to f x
.
If Fₙ
converges locally uniformly on a neighborhood of x
to a function f
which is
continuous at x
, and gₙ
tends to x
, then Fₙ (gₙ)
tends to f x
.
If Fₙ
tends locally uniformly to f
on a set s
, and gₙ
tends to x
within s
, then
Fₙ gₙ
tends to f x
if f
is continuous at x
within s
and x ∈ s
.
If Fₙ
tends uniformly to f
on a set s
, and gₙ
tends to x
within s
, then Fₙ gₙ
tends to f x
if f
is continuous at x
within s
.
If Fₙ
tends locally uniformly to f
, and gₙ
tends to x
, then Fₙ gₙ
tends to f x
.
If Fₙ
tends uniformly to f
, and gₙ
tends to x
, then Fₙ gₙ
tends to f x
.