I'm currently working as a (undergrad) TA for a mutlivariable calculus class and while covering the topic of multivariable differentiability, one of our students suggested that it'd be nice to have some actual functions we could use as examples to navigate how the theorems can be used to both establish differentiability and to flip them back around to deduce things about the partial derivatives if they're not differentiable.
In particular, we were discussing a sufficient condition for differentiability which, roughly, states that if all of the partial derivatives of a scalar field are continuous at a given point, then the field is also differentiable at that point. But more specifically I was warning them about how the contraposition of this statement does not forbid a function from being differentiable while having discontinous partial derivatives, as this beautiful example shows.
The previous example has all of its partial derivatives be discontinuous, but then a student asked if I knew any scalar fields which only had one or two discontinuous, while the rest remained continuous at the given point. Here's where I ask for your help :) I have looked around in some of the literature but haven't been able to find good enough example of this, although in principle this should be possible (from what I understand). Also, if you have other interesting examples that show how differentiability can be wrinkly or deceiving I'd really appreciate it (and our students would too)!