Wow. I remember that I thought exactly this problem out of curiosity as a high school student but did not reach an answer. In fact, I was thinking about posting this problem on MathOverflow!

At least it is easy to construct one solution: f(x)=x^φ/φ^φ+1, where φ=(1+√5)/2 is the golden ratio.

Wow. I remember that I thought exactly the same problem out of curiosity as a high school student but did not reach an answer. In fact, I was thinking about posting this problem on MathOverflow!

At least it is easy to construct one solution: f(x)=x^φ/φ^φ−1, where φ=(1+√5)/2 is the golden ratio.

Edit: Corrected the calculation. Thanks to Aaron Meyerowitz for spotting the error!