Monotonicity of odd function and 1-1 function

[xsw001]

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.

 

[arildno]

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

 

[nicksauce]

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

 

[arildno]

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

Oh dear..:shy: