Find a modulus of continuity $\delta_{\epsilon,x,y}$ for the continuous function $f(x,y)=\sqrt{1+e^{xy}}$. Find a modulus of continuity $\delta_{\epsilon,x,y}$ for the continuous function $f(x,y)=\sqrt{1+e^{xy}}$.
Side issue: since the question uses $\delta_{\epsilon,x,y}$, does that mean $\delta$ may depend on $x,y$?  So we are basically finding a modulus for an arbitrary point?
 A: Note: Depending on what the OP means by "Modulus of Continuity at a point $(x_0, y_0)$" this might not totally answer the question

We seek a function $\omega$, which we call the "modulus of continuity of $f$", satisfying $d_1(f(z_1) - f(z_2)) \le \omega( d_2(z_1,z_2) )$ for all $z_1, z_2$ in our domain. We don't actually get one here, since our function $f$ is not uniformly continuous. Nevertheless, we will derive something similar which will hopefully answer the query at hand.
We will do this over the course of a few steps. First note that given any $z_1 = (x_1, y_1)$ and $z_2 = (x_2, y_2)$ in $\mathbb{R}^2,$ and given the $f$ in your post,
\begin{align} d_1(f(z_1), f(z_2)) &= |f(z_1),f(z_2)|\\ &= |\sqrt{1+e^{x_1 y_1}}-\sqrt{1+e^{x_2 y_2}}|\\ &= \left|\frac{e^{x_1 y_1}-e^{x_2 y_2}}{\sqrt{1+e^{x_1 y_1}}+\sqrt{1+e^{x_2 y_2}}}\right|\\ &\le |e^{x_1 y_1}-e^{x_2 y_2}| \end{align}
The problem now comes down to expressing this in terms of $d_2(z_1, z_2).$ We will use a few standard analysis tricks. First, let's consider the easier case $|e^x - e^y|$ for $x, y$ in $\mathbb{R}.$ Note that the mean value theorem tells us that there is some $c \in (x,y)$ such that $|e^x - e^y|\le e^c|x-y| \le e^{\max(x,y)}|x-y|.$ If we let $x = x_1y_1$ and $y = x_2 y_2$ and do this argument we find
$$|e^{x_1 y_1}-e^{x_2 y_2}| \le e^{\max(x_1y_1,x_2y_2)}|x_1y_1-x_2y_2| $$
We now need to simplify the quantity $|x_1y_1-x_2y_2|.$ To do this, note
\begin{align}|x_1y_1-x_2y_2| &= \left|x_1y_1-x_2y_1+x_2y_1-x_2y_2\right|\\
&\le \left|x_1y_1-x_2y_1\right|+\left|x_2y_1-x_2y_2\right|\\
&= \left|y_1\right|\left|x_1-x_2\right|+\left|x_2\right|\left|y_1-y_2\right|\\
&= \max(|y_1|,|x_1|)(|x_1-x_2|+|y_1-y_2|)\end{align}
Finally, we recall that $\lVert \cdot \rVert_1 \le \sqrt{2}\lVert \cdot \rVert_2 $ on $\mathbb{R}^2,$ so that
$|x_1-x_2|+|y_1-y_2| \le \sqrt{2}d_2(z_1,z_2).$
Putting all of this together, we get the following:
\begin{align} d_1(f(z_1), f(z_2)) & \le |e^{x_1 y_1}-e^{x_2 y_2}|\\
& \le e^{\max(x_1y_1,x_2y_2)}|x_1y_1-x_2y_2|\\
& \le e^{\max(x_1y_1,x_2y_2)}\max(|y_1|,|x_1|)(|x_1-x_2|+|y_1-y_2|)\\
& \le \underbrace{\sqrt{2}e^{\max(x_1y_1,x_2y_2)}\max(|y_1|,|x_1|)d_2(z_1, z_2)}_{\omega?}\\
& \stackrel{?}{=} \omega(d_2(z_1, z_2))
\end{align}
This is of course not quite the way we want $\omega$ to work for a true modulus of continuity, but I'm not quite sure what your textbook is looking for where you say "modulus of continuity at a point"
