If \(f: \mathbb R \rightarrow \mathbb R\) is continuous everywhere and for every real \(x\),

\[ f(x) = f(2x) \]

Prove that it is indeed constant. The answer involves a bit of imagination.

If \(f: \mathbb R \rightarrow \mathbb R\) is continuous everywhere and for every real \(x\),

\[ f(x) = f(2x) \]

Prove that it is indeed constant. The answer involves a bit of imagination.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

TopNewestI think I have seen this question before.

\[f(x)=f(2x)\] \[f(\frac12x)=f(x)\] and like this, we get, \[f(\frac{x}{2^n})=f(x)\] Now as \(n\to\infty\) \[f(0)=f(x)\] and thus, we get that \(f(x)\) is a constant function. – Aditya Agarwal · 1 year, 3 months ago

Log in to reply

– Romanos Molfesis · 1 year, 3 months ago

Correct!Log in to reply

We have \(f(x)=f(2^nx)\) for all real \(x\) and for all integers \(n\). Now, by continuity, \(f(0)=\lim_{n\to -\infty}f(2^nx)=f(x)\), showing that \(f(x)\) is constant, taking the value \(f(0)\) for all \(x\). – Otto Bretscher · 1 year, 3 months ago

Log in to reply

– Romanos Molfesis · 1 year, 3 months ago

Correct!Log in to reply

it is a continuous function therfore it has to be constant – Sashank Bonda · 2 months, 1 week ago

Log in to reply