I am asked to show that there is a $2\pi$ periodic function $f$ and a $2\pi$ period function $g$ such that $f\in L_{\infty}(\Pi)$ and $g\in \Lambda^{1}(\Pi)$ ($g$ is Lipschitz) with the following properties.
(1) $\displaystyle\lim\limits_{h\rightarrow 0}||f - f(\cdot - h)||_{\infty} \neq 0$ (where $||f||_{\infty} = $ ess sup $\left\{|f(t)|: -\pi \leq t < \pi\right\}$ )
and
(2) $\displaystyle\lim\limits_{h\rightarrow 0}||g - g(\cdot - h)||_{\Lambda^{1}} \neq 0$ (where $||f||_{\Lambda^{1}} = \sup \left\{|f(t)| : -\pi \leq t < \pi\right\} + \sup \left\{\frac{|f(t + y) - f(t)|}{|y|} : -\pi \leq t < \pi, y\neq 0\right\}$ )
Note: By $f(\cdot - h)$ I mean the function which maps $x$ to $f(x-h)$. We use this in class because it is very convenient but I'm not sure if it is standard.
Part (1) is straight forward, I just took $f$ to be the step function $-\chi_{[-\pi,0)} + \chi_{[0,\pi)}$. It turned out that $||f - f(\cdot - h)||_{\infty} = -2$ for all $h > 0$, which was sufficient to establish (1).
I think the reason I am struggling with part (2) so much is that I don't really have a good understanding of what the Lipschitz "metric" is doing. With the $L_{\infty}$ norm the induced metric is simply the largest place where the functions differ. I cannot see an obvious interpretation for the induced metric from the Lipschitz norm. Any advice on where to start?