Is the Function f(x) Odd or Even? Proving with Real Numbers

[rattanjot14]

Homework Statement

1) f(x+1)=f(x)+1
2) f(x^2) =(f(x))^2
let a function real to real satisfy the above statements then prove whether the fuction is odd or even.

Homework Equations

The Attempt at a Solution

using the 2) we get 1) f(0) = 0,1
2) f(1) = 0,1
putting x = 0 in the 1st equation we get f(0) = 0 and f(1) = 1. from this we can prove f(-1) = -1 and for integers we get f(-x) = -f(x). But how to prove for real

 

[mfb]

Please do not post threads in multiple forums.

Did you consider $f(\sqrt{n}^2)$ for integers n?
What about f(1/2) and f(-1/2)?

I am not sure if it is possible to construct the whole function in that way. Based on your function values, it is trivial to show that the function cannot be even, but that is not sufficient to show that it is odd. Continuity would be nice to have.

 

[rattanjot14]

We can say it is not an even function ..then how to prove that it is an odd function.

 

[haruspex]

What can you say about |f(x)| compared with|f(-x)| on the one hand, and f(|x|) compared with f(-|x|) on the other?