4.5.2. This exercise shows that all partial derivatives of a function can exist at and about a point without being continuous at the point. Define f : R² → R by 2xy x²+y² if (x, y) # (0,0), f(x, y) if (x, y) = (0,0). (a) Show that Dıf(0,0) = D2f (0,0) = 0. (b) Show that Dif(a, b) and D2f(a, b) exist and are continuous at all other (a, b) E R². (c) Show that Dıf and D2f are discontinuous at (0,0).
4.5.2. This exercise shows that all partial derivatives of a function can exist at and about a point without being continuous at the point. Define f : R² → R by 2xy x²+y² if (x, y) # (0,0), f(x, y) if (x, y) = (0,0). (a) Show that Dıf(0,0) = D2f (0,0) = 0. (b) Show that Dif(a, b) and D2f(a, b) exist and are continuous at all other (a, b) E R². (c) Show that Dıf and D2f are discontinuous at (0,0).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Solve Problem 4.5.2
Transcribed Image Text:Exercises
4.5.1. Explain why in the discussion beginning this section the tangent
plane P consists of all points (a, b, f(a, b)) + (h, k,T(h, k)) where T(h, k) =
o' (a)h + v'(b)k.
4.5.2. This exercise shows that all partial derivatives of a function can exist at
and about a point without being continuous at the point. Define f : R?
by
> IR
2xy
x²+y²
if (x, y) # (0,0),
if (x, y) = (0,0).
f (x, y)
(a) Show that Dıf(0,0) = D2f(0,0) = 0.
(b) Show that Dif(a,b) and D2f(a,b) exist and are continuous at all other
(a, b) E R².
(c) Show that Dıf and D2f are discontinuous at (0,0).
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