tional derivative Daf. Describe precisely what the relationship is. b) In WA2 Q2(c) we saw that the existence of fa, fy and dsf is not sufficient to guarantee the differentiability of f. It might be tempting to think that differentiability will be guaranteed if all of the directional derivatives were to exist. This is not the case. In fact, this does not even guarantee continuity! Find an example of a function f : R² → R and a point (a, b) such that: (i) Daf(a, b) exists for every unit vector i E R², but (ii) f is not continuous at (a, b). Be sure to justify vour claims.
tional derivative Daf. Describe precisely what the relationship is. b) In WA2 Q2(c) we saw that the existence of fa, fy and dsf is not sufficient to guarantee the differentiability of f. It might be tempting to think that differentiability will be guaranteed if all of the directional derivatives were to exist. This is not the case. In fact, this does not even guarantee continuity! Find an example of a function f : R² → R and a point (a, b) such that: (i) Daf(a, b) exists for every unit vector i E R², but (ii) f is not continuous at (a, b). Be sure to justify vour claims.
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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Transcribed Image Text:(a) The simultaneous derivative df defined in WA2 Q2 is very closely related to the direc-
tional derivative Duf. Describe precisely what the relationship is.
(b) In WA2 Q2(c) we saw that the existence of fr, fy and dgf is not sufficient to guarantee
the differentiability of f. It might be tempting to think that differentiability will be
guaranteed if all of the directional derivatives were to exist. This is not the case. In
fact, this does not even guarantee continuity!
Find an example of a function f: R² → R and a point (a, b) such that:
(i) Dūf(a, b) exists for every unit vector i e R2, but
(ii) f is not continuous at (a, b).
Be sure to justify your claims.
(Tint: An ovomnlo of cuoh
f hoo onnoorod comouhoro in WA1
21
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