Results 1 to 11 of 11

Math Help - How do I prove that {x: f(x) not eqaul to r_0} is an open set?

  1. #1
    Member
    Joined
    Feb 2010
    Posts
    146

    How do I prove that {x: f(x) not eqaul to r_0} is an open set?

    Hey guys, how do i solve this problem?

    Suppose f: R->R is continuous and let r_0 \inR. Prove that {x: f(x) not equal to r_0 is an open set.

    My second midterm is coming up and i'm trying to study for it but i'm having trouble with some of the problems. i feel that when I see it i can learn it. Thanks guys (ps there will probably a few, considering how many questions are on this review sheet )
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by tn11631 View Post
    Hey guys, how do i solve this problem?

    Suppose f: R->R is continuous and let r_0 \inR. Prove that {x: f(x) not equal to r_0 is an open set.

    My second midterm is coming up and i'm trying to study for it but i'm having trouble with some of the problems. i feel that when I see it i can learn it. Thanks guys (ps there will probably a few, considering how many questions are on this review sheet )
    What is the inverse image of an open set under a continuous map?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Feb 2010
    Posts
    146
    Quote Originally Posted by Drexel28 View Post
    What is the inverse image of an open set under a continuous map?
    as in the inverse image of every open set under f is again open?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by tn11631 View Post
    as in the inverse image of every open set under f is again open?
    Bingo.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Feb 2010
    Posts
    146
    hm, I don't recall going in this direction in class. Sorry as you can tell this isn't my favorite math. Do I have to show that for r_0 in f, f is a neighborhood of r_0? Sorry once I learn one proof I'm able to push myself through similar ones, but i just completely draw blanks.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by tn11631 View Post
    hm, I don't recall going in this direction in class. Sorry as you can tell this isn't my favorite math. Do I have to show that for r_0 in f, f is a neighborhood of r_0? Sorry once I learn one proof I'm able to push myself through similar ones, but i just completely draw blanks.
    I mean, I'm sorry to report that I am not a mind-reader. I have no idea what definition of continuity you are given. That would be a nice start.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Feb 2010
    Posts
    146
    Quote Originally Posted by Drexel28 View Post
    I mean, I'm sorry to report that I am not a mind-reader. I have no idea what definition of continuity you are given. That would be a nice start.
    I apologize if i have upset you or offended you in anyway those weren't my intentions i'm just lost. However the definition the books gives is: "Suppose E \subsetR and f: E->R. If x_0 \inE, then f is continuous at x_0 iff each \epsilon>0, there is a delta>0 such that if
    |x- x_0|<delta, x \inE, then |f(x)-f( x_0)< \epsilon. If f is continuous at x for every x \inE, then we say f is continuous"
    I don't know if that gives you any help as to where i'm at but i hope it helps. thanks again
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by tn11631 View Post
    I apologize if i have upset you or offended you in anyway those weren't my intentions i'm just lost. However the definition the books gives is: "Suppose E \subsetR and f: E->R. If x_0 \inE, then f is continuous at x_0 iff each \epsilon>0, there is a delta>0 such that if
    |x- x_0|<delta, x \inE, then |f(x)-f( x_0)< \epsilon. If f is continuous at x for every x \inE, then we say f is continuous"
    I don't know if that gives you any help as to where i'm at but i hope it helps. thanks again
    So the basic concept is this. Let x\in f^{-1}(\mathbb{R}-\{0\}). Then, f(x)\in\mathbb{R}-\{0\}. So, since \mathbb{R}-\{0\} is open there exists some B_{\varepsilon}(f(x))\subseteq\mathbb{R}-\{0\}. But, by continuity there exists a \delta>0 such that \left(d(x,y)<\delta\implies d(f(x),f(y))<\varepsilon\right)\Longleftrightarrow  \left(f(B_{\delta}(x))\subseteq B_{\varepsilon}(f(x))\right). So, f(B_{\delta}(x))\subseteq\mathbb{R}-\{0\}\implies B_{\delta}(x)\subseteq f^{-1}(\mathbb{R}-\{0\}). It follows that x\in f^{-1}\left(\mathbb{R}-\{0\}\right)\implies x\in\left[f^{-1}\left(\mathbb{R}-\{0\}\right)\right]^{\circ}. Thus, f^{-1}(\mathbb{R}-\{0\}) is open.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    Feb 2010
    Posts
    146
    Quote Originally Posted by Drexel28 View Post
    So the basic concept is this. Let x\in f^{-1}(\mathbb{R}-\{0\}). Then, f(x)\in\mathbb{R}-\{0\}. So, since \mathbb{R}-\{0\} is open there exists some B_{\varepsilon}(f(x))\subseteq\mathbb{R}-\{0\}. But, by continuity there exists a \delta>0 such that \left(d(x,y)<\delta\implies d(f(x),f(y))<\varepsilon\right)\Longleftrightarrow  \left(f(B_{\delta}(x))\subseteq B_{\varepsilon}(f(x))\right). So, f(B_{\delta}(x))\subseteq\mathbb{R}-\{0\}\implies B_{\delta}(x)\subseteq f^{-1}(\mathbb{R}-\{0\}). It follows that x\in f^{-1}\left(\mathbb{R}-\{0\}\right)\implies x\in\left[f^{-1}\left(\mathbb{R}-\{0\}\right)\right]^{\circ}. Thus, f^{-1}(\mathbb{R}-\{0\}) is open.
    hmm. Ok I deff see where and how this proof is working. However I have what i think is a stupid question. did you take the inverse image b/c of the condition r_0 \inR and having to show that f(x) \neq r_0. And also in like the second to last thing, that's the compliment right? I just wanna make sure sorry.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by tn11631 View Post
    hmm. Ok I deff see where and how this proof is working. However I have what i think is a stupid question. did you take the inverse image b/c of the condition r_0 \inR and having to show that f(x) \neq r_0.
    I don't know what you mean.

    And also in like the second to last thing, that's the compliment right? I just wanna make sure sorry.
    When is something every in a set and it's compliment? It stand for interior.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Member
    Joined
    Feb 2010
    Posts
    146
    Sorry, I was refering back to the original question. I think i'm ok now. Thanks
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. prove that u is open
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: December 3rd 2011, 07:19 PM
  2. Replies: 1
    Last Post: October 30th 2010, 01:50 PM
  3. Prove the following Set is Open
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 2nd 2010, 07:50 AM
  4. Prove: Interior is an open set
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: February 20th 2010, 01:02 PM
  5. Prove that a set is open.
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 13th 2009, 11:01 AM

Search Tags


/mathhelpforum @mathhelpforum