Let $g:\mathbb{R}^n \rightarrow \mathbb{R}$, let $t \in \mathbb{R}$, suppose $s \in [0, t]$ and let $e_i$ denote the standard $i^{th}$ basis vector. I have read the following claim:
$ \frac{1}{t} \int^t_0 |g(x + se_i) - g(x)|ds \leq \max_{0 \leq s \leq t}|g(x + se_i) - g(x)| $
How can I see that this claim is true? I really have no thoughts as to how one might proceed to show this except, perhaps, to convert the LHS to a Riemann sum and manipulate that to show that it is less than the RHS. But this does not seem to be a very good approach.