Meaning of pointwise and uniformly in mathematics?

[teng125]

can anybody pls explain to me what is the meaning of pointwise and uniformly in mathematics??
i really don't know what is that mean...thanx...

 

[HallsofIvy]

"pointwise" convergence and "uniform" convergence of sequences of functions.

The sequence of functions {f_n} converges to the function f pointwise if the sequences of numbers {f_n(x)} converges to the number f(x) for every value of x (every point).
You might recall that that requires that "for every \epsilon> 0, there exist \delta>0 so that if |x- x_0|<\delta, then |f(x)- f(x_0)|< \epsilon. The choice of \delta may depend on both \epsilon and x₀.

The sequence of functions {f_n} converges uniformly if, for a given \epsilon, you can choose a single \delta that will work for any x₀ in the set.

It's trival to prove that if a sequence of functions converges uniformly to a function, then the sequence converges pointwise to the same function.

It's much harder to prove that if a sequence of functions converges pointwise to an function, on a compact (closed and bounded) set, then the sequence converges uniformly to that same function.

You can do a similar thing to define "pointwise" continuous and "uniformly continuous" on a set.

 

[teng125]

what is the meaning of a "sequence of functions converges uniformly to a function"?? is it means the function will exists as a number=1 or 0??