Directional derivative of $F(x, y)=\sum_{0\leq i, j\leq m}z_{i,j} \max(1-|i-mx|)\max(1-|j-my|)$

Question

Let $F:[0,1]\times [0,1]\to [0,1]$ be the following continuous function.

$$F(x, y)=\sum_{0\leq i, j\leq m}z_{i,j} \max(1-|i-mx|)\max(1-|j-my|)$$

I am trying to calculate

$$\lim\sup_{\epsilon \to 0+} \frac{F(x+\epsilon, y-\epsilon)-F(x, y)}{\epsilon}$$

When $x \in (\frac{i_0}{m}, \frac{i_0+1}{m})$ and $y\in (\frac{j_0}{m}, \frac{j_0+1}{m})$ for some integer $i_0, j_0$, then the function is differentiable at that point, and this just reduces to a directional derivative. However, I'm not sure how to handle when $x=i_0/m$ or $y=j_0/m$. I might be missing something obvious.

Answer 1

Directional derivative is a linear operator and the multipliers $z_{i,j}$ are not essential. Therefore, it is enough to consider (with Mathematica)

ResourceFunction["DirectionalDerivative"][Max[1-RealAbs[i-m x]]*Max[1 - RealAbs[j - m y]], 
{x, y}, {a, b}, {Cos[\[Alpha]], Sin[\[Alpha]]}]

$ \frac{m \cos (\alpha ) (i-a m) (1-| j-b m| )}{| i-a m| \sqrt{| \sin (\alpha )| ^2+| \cos (\alpha )| ^2}}+\frac{m \sin (\alpha ) (j-b m) (1-| i-a m| )}{\sqrt{| \sin (\alpha )| ^2+| \cos (\alpha )| ^2} | j-b m| }$

and

ResourceFunction["DirectionalDerivative"][Max[1-RealAbs[i - m x]]*Max[1- RealAbs[j - m y]], 
{x, y}, {i/m, j/m}, {Cos[\[Alpha]], Sin[\[Alpha]]}]

$\text{Indeterminate}$

Addition.

MaxLimit[((1 - RealAbs[j - m (j/m - \[Epsilon])]) (1 - 
   RealAbs[i - m (i/m + \[Epsilon])]) - 
1)/\[Epsilon], \[Epsilon] -> 0, Direction -> "FromAbove"]

$-2 \sqrt{m^2} $