6-152 Find all functions $f$ from the set of real numbers to the set of real numbers that satisfy the following conditions:
(1) $f(x)$ is strictly increasing;
(2) For all real numbers $x, f(x) + g(x) = 2x$, where $g(x)$ is the inverse function of $f(x)$.
[Solution] We prove that $f(x)-x=c$, where $c=f(0)$.
In fact, if we let $S_{c}=\{x \mid f(x)-x=c\}$, then $0 \in S_{c}$, so $S_{c}$ is non-empty.
If $a \in S_{c}$, then $f(a)=a+c$, hence $g(a+c)=a$, $f(a+c)=2(a+c)-a=(a+c)+c$, which means $a+c \in S_{c}$.

Suppose $c' \neq c$ and $S_{c'}$ is not empty, then the proof above indicates that $S_{c'}$ would be empty, a contradiction. Therefore, for all $c' \neq c$, $S_{c'}$ is empty. In other words, $S_{c}$ contains all real numbers, thus, $f(x)=x+c$.