Chapter 4: Partial Differentiation
Section 4.11: Differentiability
Show that the function fx,y in Table 4.11.1 has a differential at the origin, and hence, is differentiable at the origin.
For f to be differentiable at the origin, Δ f≡f0+h,0+k−f0,0=fh,k must assume the form
fx0,0 h+fy0,0 k+ηh,k⋅h2+k2
where η→0 as h,k→0,0. Since fx0,0=fy0,0=0 from Example 4.11.1, it follows that
where λx,y=h2+k2⁢sin1h2+k2. Since λ is the product of a bounded factor and a factor that goes to zero, λ→0 as h,k→0,0. Hence, setting η=λ implies that f is differentiable at the origin.
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