Let $f \colon [0,1] \to \mathbb{R}$ be a function and define $$\varpi'_f(\delta) = \inf_{\{t_i\}} \max_{i=1,\dots,n} \sup_{t,s \in [t_{i-1},t_{i})}|f(t)-f(s)|$$ where the infimum is taken over all partitions $0=t_0<t_1 < \cdots <t_n =1$ with $\min_{i=1,\dots,n}(t_i - t_{i-1})> \delta$.
I was trying to prove something like $$\varpi'_{f+g}(\delta) \leq \varpi'_f(\delta) + \varpi'_g(\delta)$$ but I wasn't be able to do it because of the infimus
any help will be appreciated
This is not true. Take:
\begin{align} &f(t) = \left\{ \begin{array}{ll} 0 &\mbox{ if $0 \leq x < 0.5$} \\ 1 & \mbox{ if $0.5 \leq x \leq 1$} \end{array} \right.\\ &g(t) = \left\{ \begin{array}{ll} 0 &\mbox{ if $0 \leq x < 0.6$} \\ 1 & \mbox{ if $0.6 \leq x \leq 1$} \end{array} \right.\\ \end{align}
Then $\varpi_f'(0.3) = \varpi_g'(0.3) = 0$ but $\varpi_{f+g}'(0.3)=1$.