Uniform Convergence And Integration

Integral Convergence

Let

(f_{n}) : ℕ_{1} \to C [a, b]

such that

f_{n} \overset{u n i f}{\to} f

and fix

c \in [a, b]

then the new sequence of functions

F_{n} : [a, b] \to ℝ

F_{n} (x) : = \int_{c}^{x} f_{n} (t) d t

converges uniformly on

[a, b]

to the function

F (x) : = \int_{c}^{x} f (t) d t

Derivative Convergence

Suppose that

f_{n}

is a sequence of continuously differentiable functions on

[a, b]

such that

f_{n}^{'}

converges uniformly to a function

g

and there is a point

c \in [a, b]

such that

\lim_{n \to \infty} f_{n} (c) = γ

exists. Then

f_{n}

converges uniformly to a differentiable function

f

with

f (c) = γ

and

f^{'} = g