# Meaning of pointwise and uniformly in mathematics?

1. Jan 19, 2006

### 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....

2. Jan 19, 2006

### HallsofIvy

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

The sequence of functions {fn} converges to the function f pointwise if the sequences of numbers {fn(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 {fn} 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.

3. Jan 19, 2006

### 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??