Cauchy's Functional Equation Revisited

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

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

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.

#Algebra

Easy Math Editor

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Comments

Wikipedia :P

A Former Brilliant Member - 5 years ago

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

Sharky Kesa - 5 years ago

Anyone can edit wikipedia :v

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

Because you guys are more trusted than wikipedia

Jason Chrysoprase - 5 years ago

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

A Former Brilliant Member - 5 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$