Locally monotone Boolean function
An example of a locally monotone function is $g(x,y) = x \cdot (1-y)$. In this example $g$ is monotone increasing in $x$ and monotone decreasing in $y$.

An example of a function that is not locally monotone is $h(x,y) = x \oplus y$.
Let's see why it is not locally monotone. Look at the $x$ variable. (1) It is neither monotone increasing in $x$ since $h(0,1) &gt; h(1,1)$, (2) nor is it monotone decreasing in $x$ since $h(0,0) &lt; h(1,0)$.

Locally monotone functions are called also unate functions.