[color=darkred]Find all functions $f:\mathbb{R}\to\mathbb{R}$ with the following property: for any open bounded interval $I$, the set $f(I)$ is an open interval having the same length with $I$ .[/color]
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   Removing the restriction \( f(0) = 0 \), the general solution is:
   \[
   f(x) = \pm x + C
   \]
   where \( C \) is a constant.

The final answer is \( \boxed{ f(x) = \pm x + C } \).