Pointwise and uniform limits Continuous function

a sequence of continuous functions fn(x) (pointwise) limit function f(x) discontinuous. convergence not uniform.

given sequence

f

1


,

f

2


,
…
:
i
→

r



{\displaystyle f_{1},f_{2},\dotsc \colon i\rightarrow \mathbf {r} }

of functions such limit

f
(
x
)
:=

lim

n
→
∞



f

n


(
x
)


{\displaystyle f(x):=\lim _{n\rightarrow \infty }f_{n}(x)}

exists x in d, resulting function f(x) referred pointwise limit of sequence of functions (fn)n∈n. pointwise limit function need not continuous, if functions fn continuous, animation @ right shows. however, f continuous when sequence converges uniformly, uniform convergence theorem. theorem can used show exponential functions, logarithms, square root function, trigonometric functions continuous.