# about mapping, onto, and set

i know it is easy, but i am having some trouble with the basic theories.

if i am going to construct an example of a continuous function

f:(0,1) -> R

which maps the open set (0,1) onto a closed set.

i understand wt a open set, a closed set is, but once the question come altogether, i am lost.

beside, could anyone give me a simple example to explain "onto" and "mapping"?

thx

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Let

h(x) = 2* x * pi - pi

g(x) = tan(x)

f(x)= g(h(x))

It is easy to see h(x) is an 1-1, onto and conti mapping from (0,1) to ( pi, - pi ) and

tan(x) is 1-1 , onto and conti on ( pi, - pi ).

So f(g(x)) is 1-1, onto and conti on (0,1).

Since

lim (x->1-) f = oo

lim (x->0+) = - oo

We have f maps (0,1)-> (oo,-oo) = R.

By def, R is close.(R is both open and close)

mapping means a function.

f is onto means the range of f = the codomain of f

2008-09-19 12:29:12 補充：

So f(g(x)) is 1-1 and conti on (0,1).

2008-09-19 12:29:55 補充：

tan(x) is 1-1 and conti on ( pi, - pi ).

2008-09-19 12:33:50 補充：

For briefly, You can say tan(x) is 1-1 , onto and conti from (-pi,pi) to (-oo,oo) imlies

g(h(x)) is 1-1, onto and conti from (0,1) to (-oo,oo) = R.

2008-09-19 12:34:16 補充：

So g(h(x))) is 1-1 and conti on (0,1).

2008-09-19 17:18:35 補充：

1. R is both open and close.

2. f is conti implies f^(-1)(U) is open for all open set U

so the image of f may be anything.

2008-09-19 17:23:12 補充：

1. R is both open and close.

2. f is conti implies f^(-1)(U) is open for all open set U

so the image of f may be anything.

你是說 f 的映像 (image) 是 closed set in R 吧?

這很簡單，你不要找單調的函數，而是要找在 (0, 1) 內都有 maximum point 及 minimum point 的函數。

所以，就定義 f: (0, 1) → R 為　　f(x) = sin (2πx)

明顯地，f 的 image 就是 [-1, 1], a closed subset of R. 因為 -1 及 1 為 f 在 (0, 1) 內的極小與極大值！！ 再由 f 為連續，從中值定理得知 f maps (0, 1) onto [-1, 1].

2008-09-19 17:43:29 補充：

其實，我只是想舉一個 non-trivial 的例子， f 是常函數，則 f 的 image 為單點集，是一個 R 的閉子集。這問題有什麼困難的？

thx, it explains sth.

so i understand one to one means the function, one x only have one solution.

so like f(x)=2x is one to one.

f(x)=x^2 is not, because both x=2, and -2 gives the same solution.

2008-09-19 13:13:45 補充：

and on to means the solution appear and cover the space.

like f(x)=2x is on to, because all value of x cover the space.

f(x)=x^2 is not onto, because it only cover the >0 side. doesnt cover <0.

2008-09-19 13:13:54 補充：

so with my original question.

with f:(0,1) ->R

what is an example to map the open set (0,1) onto a closed set?

the examples u gave me are useful, i have a better picture to understand onto and one to one. but it is an open set. how could that be mapped to a closed set?

thx again