Results 1 to 8 of 8

Math Help - strictly monotonic

  1. #1
    Junior Member
    Joined
    Dec 2009
    Posts
    30

    Wink strictly monotonic

    Let K be a subset of R
    A function F: K--> R is said to be strictly monotonic if it is either strictly increasing or strictly decreasing. That is, one of the following 2 holds:

    1.) x,y are elements in K with x<y implies that f(x)<f(y)
    or
    2.) x,y are elements in K with x<y implies that f(x)>f(y)


    Let I be an interval in R and let f: I-->R be a continuous function.
    Prove that f is one-to-one iff f is strictly monotonic.

    I am having trouble starting this problem and any help would be much appreciated.
    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 derek walcott View Post
    Let K be a subset of R
    A function F: K--> R is said to be strictly monotonic if it is either strictly increasing or strictly decreasing. That is, one of the following 2 holds:

    1.) x,y are elements in K with x<y implies that f(x)<f(y)
    or
    2.) x,y are elements in K with x<y implies that f(x)>f(y)


    Let I be an interval in R and let f: I-->R be a continuous function.
    Prove that f is one-to-one iff f is strictly monotonic.

    I am having trouble starting this problem and any help would be much appreciated.
    Suppose that f was injective and continuous. Assume for a second that f was not monotonic, then what would the IVT give us as a contradiction?

    Conversely, if x\ne y then either x<y\implies f(x)<f(y)\implies f(x)\ne f(y) or y<x\implies f(y)<f(x)\implies f(y)\ne f(x), either way we have injectivity.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Dec 2009
    Posts
    30
    so proving that it is an injection proves that it is also one-to-one?
    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 derek walcott View Post
    so proving that it is an injection proves that it is also one-to-one?
    Injection means one-to-one.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Dec 2009
    Posts
    30
    okay that's what i thought.

    and it's only injective because f is continuous. if f were not continuous then x=y and f(x) ≠ f(y) could occur?
    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 derek walcott View Post
    okay that's what i thought.

    and it's only injective because f is continuous. if f were not continuous then x=y and f(x) ≠ f(y) could occur?
    Wait, what?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Dec 2009
    Posts
    30
    i am trying to think of the implications of f:K-->R being continuous and not being continuous

    how does the problem change? would that prove that it isn't monotonic?
    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 derek walcott View Post
    i am trying to think of the implications of f:K-->R being continuous and not being continuous

    how does the problem change? would that prove that it isn't monotonic?
    If the function is continuous and injective it is monotonic. If a function is monotonic it is automatically injective.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove that f is strictly increasing
    Posted in the Calculus Forum
    Replies: 3
    Last Post: January 18th 2010, 09:29 AM
  2. strictly decreasing
    Posted in the Algebra Forum
    Replies: 2
    Last Post: December 9th 2009, 07:35 PM
  3. strictly monotonic
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 19th 2009, 08:38 AM
  4. strictly increasing function
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: March 29th 2009, 02:40 AM
  5. Replies: 1
    Last Post: September 26th 2007, 11:01 AM

Search Tags


/mathhelpforum @mathhelpforum