Find $f'(1)$ if $f$ is continuous and such that $f (f (x))=1+x$ for every $x$ 
If $ f $ is a continuous function satisfying $f (f (x))=1+x$ find $f'(1)$.

I just guessed $f (x)=x+1/2$.Any formal method for this sum ?

The relationship $f(f(x))=x+1$ implies that $f^{-1}(x+1)=f(x)$. This implies that when the inverse function is shifted to the left by one unit it must give back the original function. Indeed, $f(x)=x+1/2$ satisfies this requirement and has $f^{-1}(x)=x-\frac{1}{2}$. Although this doesn't show that this is the only function, there may be more
