The first partial derivatives of , obtained in Example 4.11.1, are
and
To show that is not bounded, examine its limit as . The first term in is the product of with a factor that exhibits bounded oscillations. Since , this term goes to zero in the limit. The second term is the product of a term that exhibits bounded oscillations multiplied by . Since this factor, even on is the unbounded , it should be clear that becomes unbounded as .
A similar analysis, mutatis mutandis, shows that becomes unbounded as .
Given that both and become unbounded at the origin, no further argument needs to be made that indeed, these first partials are discontinuous at the origin.