A Noncontinuous Limit of a Sequence of Continuous Functions

n

16

show limit

Consider a sequence

f

n

of continuous real-valued functions of a real variable. The sequence

f

n

converges pointwise on a set

E

to a function

f

if for each

x

in

E

,

f

n

(x)f(x)

as

n∞

. The limit

f

is not guaranteed to be continuous; in this Demonstration the limit has a removable discontinuity. (To construct a limit that is discontinuous everywhere in

[0,1]

, construct

f

n

with spikes at all numbers that can be written in the form

p/q

, where

p

and

q

are positive integers and

p<q≤n

.)

The limit of a uniformly convergent sequence of continuous functions is guaranteed to be continuous. Here "uniformly" means that in the ϵ-

N

definition of the limit, the same

N

must apply to every

x

in

E

. Pointwise convergence only requires an

N

that may depend on

x

.