
Small Lipschitz space
Let Lip_m[a,b] be the set of all Lipschitz functions of order m in [a,b] that is all functions satisfying f(x)f(y)<= K * xy^m for a certain constant K and for all x ,y in [a,b].
Now define the following function:
A_m (f) = sup { f(x)  f(y)/xy^m : x,y are in [a,b] and x is not equal to y}. sup represents supremum.
Now for z >0 and f in Lip_m[a,b] and t>0 define:
R_z (f,t) = sup { f(x) f(y) / xy^z : x,y are in [a,b], 0< x y <= t}.
Now another definition:
The "small Lipschitz" space denoted by lip_m[a,b] is the following set:
lip_m[a,b] = { f in Lip_m[a,b] : limit ( R_m (f,t) when t> +0) = 0 } where t>+0 means from the right.
If m = 1 show that lip_m [a,b] contains only the constant functions.
Can you please help?

Choose $\displaystyle \epsilon>0$. Since the limit of $\displaystyle R_1(f,t)$ is zero for $\displaystyle t\rightarrow 0+$, we get some $\displaystyle \delta>0$ such that $\displaystyle f(x)f(y)<\epsilonxy$ for $\displaystyle xy<\delta$. Consider a covering of $\displaystyle [a,b]$ consisting of intervals of length $\displaystyle \delta$. Choose $\displaystyle x\in [a,b]$ and show that $\displaystyle f(a)f(x)<\epsilon$ by adding and substracting inside the absolute value.