It is commonly known that the solution for Cauchy's Functional Equation

\[ f:\mathbb{Q} \rightarrow \mathbb{Q}, \quad f(x)+f(y)=f(x+y)\]

has solutions \(f(x)=cx\). (If you don't know how to prove this, I ask for you to attempt to prove this)

But now, if we instead change this function from being in the rationals to being the reals, we get a strange property.

Let \(f\) be a function \(f:\mathbb{R} \to \mathbb{R}\) such that

\[\begin{align} f(x) + f(y) &= f(x+y) \forall x, y\\ f(1) &= 1\\ \text{ The function is not }f(x) & = x \end{align}\]

Consider the graph of \( y = f(x) \) on the plane. Prove that for any disc, no matter how small, there always exists a point that lies on the graph.

For this proof, you must accept the Axiom of Choice as true.

This question came from a corollary I used in APMO 2016 Q5.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestWikipedia :P

Log in to reply

How do you know Wikipedia is correct? Anyone can edit it.

Log in to reply

Anyone can edit wikipedia :v

For math homework, i usually go here if i'm confused

Because you guys are more trusted than wikipedia

Log in to reply

I've seen their proof. It was right in my opinion.

Log in to reply

Comment deleted Jun 04, 2016

Log in to reply

Comment deleted Jun 04, 2016

Log in to reply

Comment deleted Jun 04, 2016

Log in to reply

Log in to reply

Log in to reply