Function with minimum sensitivity
The tribes function is the classical example of a transitive balanced function whose average sensitivity is $\Theta(\log n)$. Its maximal sensitivity is however $\Theta(n/\log n)$.

The addressing function is an example of a balanced function depending on all inputs which has sensitivity $\Theta(\log n)$ and average sensitivity $\Theta(\log n)$. It is simpler to describe this function as having an input of size $n + \log n$: the first $n$ bits describe a binary string $x$, and the last $\log n$ bits describe a number $i \in [1,n]$. The output of the function is $x_i$. At any point the function depends only on the $\log n$ bits of $i$ as well as the single bit $x_i$.