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Meaning of pointwise and uniformly in mathematics?

  1. Jan 19, 2006 #1
    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. jcsd
  3. Jan 19, 2006 #2

    HallsofIvy

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    "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 [itex]\epsilon> 0[/itex], there exist [itex]\delta>0[/itex] so that if [itex]|x- x_0|<\delta[/itex], then [itex]|f(x)- f(x_0)|< \epsilon[/itex]. The choice of [itex]\delta[/itex] may depend on both [itex]\epsilon[/itex] and x0.

    The sequence of functions {fn} converges uniformly if, for a given [itex]\epsilon[/itex], you can choose a single [itex]\delta[/itex] 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.
     
  4. Jan 19, 2006 #3
    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??
     
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