Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Monotonicity of odd function and 1-1 function

  1. Oct 23, 2010 #1
    I have two general questions that I'm NOT sure if it's absolutely accurate statement or NOT.

    1) Odd function is always strictly monotone, either strictly increasing or strictly decreasing right? If there any counterexample to disprove my assumption?

    2) One-to-one function is always strictly monotone right? Is there any counterexample to disprove my assumption?

    Note: This is NOT homework questions. These are my general assumptions.
     
  2. jcsd
  3. Oct 23, 2010 #2

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    1) Incorrect.
    For example, the function f(-1)=1, f(1)=-1, f(x)=0 otherwise is neither strictly increasing or decreasing.
    Making a continuous function of this one, with "humps" around x=-1 and 1 is another simple counter-example.

    F(x)=0 is a third counterexample.

    2) Correct
     
  4. Oct 23, 2010 #3

    nicksauce

    User Avatar
    Science Advisor
    Homework Helper

    1) an obvious counterexample is f(x) = sin(x)

    2) should be false, if you don't impose that the function has to be continuous

    E.g.:
    f(x) = x {x != 2,3}
    f(2) = 3
    f(3) = 2
     
  5. Oct 23, 2010 #4

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    Oh dear..:shy:
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook