Find $f$ such that $\{f(x+y)\}=\{f(x)\}+\{f(y)\}$

Question

Find all continuous function such that $\{f(x+y)\}=\{f(x)\}+\{f(y)\}$ for all $x, y\in\mathbb{R}$. Denote $\{x\}=x-[x]$ in which $[x]$ is the largest integer number does not exceed x.

Answer 1

For all $x \in \mathbb{R}$, we have $0 \leq \{x\} <1$.

Suppose $\{f(x+y)\} = \{f(x)\} + \{f(y)\}$. Can you show that there cannot be any $x$ such that $\{f(x)\} \geq 0.5$?

Can you show that there cannot be any $x$ such that $\{f(x)\}>0$?

Answer 2

If $f$ ever crosses an integer boundary, say at $x$, there's a jump by $\pm1$ in $\{f(x)\}$. Then for given $y$, there would have to be a corresponding jump in $\{f(x+y)\}$. But since we can choose $y$ arbitrarily, that would mean that $f$ would have to cross an integer boundary at every point, which is impossible.

Thus $f$ never crosses an integer boundary. But then $g(x)=\{f(x)\}$ is a continuous function with $g(x+y)=g(x)+g(y)$. An additive continuous function must be linear; but the only linear function that can be the fractional part of $f$ is the zero function. Thus $f$ takes a constant integer value.