Suppose for $f: [0,1] \to R$, we have $|f(x) - f(y) | \le K |x-y|$, and $f(0)= f(1) = 0$. Prove that $|f(x)| \le K/2$. Further show by example that $K/2$ is the best possible, that is, there exists such a continuous function for which $|f(x)| = K/2$ for some $x \in [0 ,1]$.
I am trying to find $x$ and $y$ that satisfy $|f(x)| \le K/2$, but I constantly fails. Can you give some help? I also appreciate if you give some hint for the second part of the question (Further show ~).