(a) Let f(x, y) be defined on (a, b) × (c,d).(i) Give an example to show that existence of partial derivatives f(x, y) andfy(x, y) does not imply continuity of f(x, y) at (x, y).(ii) Show that if f is differentiable at (x, y), then it is continuous there.(c) State and prove the Clairaut Theorem.(d) Give one example to illustrate part (c), and justify your answer.

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Answered: (a) Let f(x, y) be defined on (a, b) ×…

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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(a) Let f(x, y) be defined on (a, b) × (c,d). (i) Give an example to show that existence of partial derivatives fx(x, y) and fy(x, y) does not imply continuity of f(x, y) at (x, y). (ii) Show that if f is differentiable at (x, y), then it is continuous there.