real analysis - Pathological Question involving $C^1$ Criterion for Differentiability

Edwards1973 gives a sufficient condition for differentiability:

If all partial derivatives of $f$ exist at every point of an open set
containing $\vec a$, and the partials are continuous at $\vec a$, then
$f$ is differentiable at $\vec a$.

I am wondering if the first condition, existence of all partials in an open set containing $\vec a$, is needed. Is the second condition alone, continuity of partials at $\vec a$, sufficient for differentiability?

Another way to phrase this (but in the opposite sense):
Does there exist a function $f$ satisfying these conditions:
1. $f$ is not differentiable at $\vec a$,
2. all of its partial derivatives are continuous at $\vec a$, and
3. one or more of the partials are not defined in some parts of any open set containing $\vec a$.