A typical realization of $W_1(s)+W_2(t)$, where $W_1$ and $W_2$ are standard independent Wiener processes, defines a function with requested properties.

Of course, there are simpler examples.

UPD. The following example has no probability in it: take any function $g:\mathbb{R}\to\mathbb{R}$ that has upper one-sided derivatives at each point equal to $+\infty$ and lower ones equal to $-\infty$ and set $f(x,y)=g(x)+g(y)$.

A typical realization of $W_1(s)+W_2(t)$, where $W_1$ and $W_2$ are standard independent Wiener processes, defines a function with requested properties.

Of course, there are simpler examples.

UPD. The following example has no probability in it: take any continuous function $g:\mathbb{R}\to\mathbb{R}$ that has upper one-sided derivatives at each point equal to $+\infty$ and set $f(x,y)=g(x)+g(y)$.

A typical realization of $W_1(s)+W_2(t)$, where $W_1$ and $W_2$ are standard independent Wiener processes, defines a function with requested properties.

Of course, there are simpler examples.

A typical realization of $W_1(s)+W_2(t)$, where $W_1$ and $W_2$ are standard independent Wiener processes, defines a function with requested properties.

Of course, there are simpler examples.

UPD. The following example has no probability in it: take any function $g:\mathbb{R}\to\mathbb{R}$ that has upper one-sided derivatives at each point equal to $+\infty$ and lower ones equal to $-\infty$ and set $f(x,y)=g(x)+g(y)$.