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

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Homework Help Overview

The discussion revolves around determining whether a function defined by specific properties is odd or even. The properties include functional equations involving f(x+1) and f(x^2), with the goal of proving the nature of the function based on these equations.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants explore the implications of the functional equations, questioning how the function behaves at specific points such as f(0), f(1), and f(-1). There are inquiries about the function's behavior for non-integer values and considerations of continuity.

Discussion Status

Some participants have noted that the function cannot be even, while others are seeking ways to establish whether it is odd. There is an ongoing exploration of various function values and their relationships, with no explicit consensus reached yet.

Contextual Notes

Participants are considering the implications of the function's values and the potential need for continuity in the proof. There is also a mention of constraints regarding posting in multiple forums.

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

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