Chapter 4: Partial Differentiation
Section 4.11: Differentiability
Show that the first partial derivatives of the function gx,y in Table 4.11.1 are not bounded, and are not continuous.
The first partial derivatives of g, obtained in Example 4.11.1, are
To show that gxx,y is not bounded, examine its limit as x,y→0,0. The first term in gx is the product of x with a factor that exhibits bounded oscillations. Since x→0, this term goes to zero in the limit. The second term is the product of a term that exhibits bounded oscillations multiplied by x/x2+y2. Since this factor, even on y=0 is the unbounded 1/x, it should be clear that gxx,y becomes unbounded as x,y→0,0.
A similar analysis, mutatis mutandis, shows that gyx,y becomes unbounded as x,y→0,0.
Given that both gx and gy become unbounded at the origin, no further argument needs to be made that indeed, these first partials are discontinuous at the origin.
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