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1. (a) Define f: R² → R by sin(r-y) f(x, y):= if y #x, f(t, t) = 1 Vt ER. Prove that f is continuous at all points of R?. (b) Define g: R2 → R by sin(ry) r² + y2" (x, y) E R? \ {(0,0)}, g(x, y) g(0,0) = 0. %3D Prove that g is separately continuous, but not continuous, at (0,0).

1. (a) Define f: R² → R by sin(r-y) f(x, y):= if y #x, f(t, t) = 1 Vt ER. Prove that f is continuous at all points of R?. (b) Define g: R2 → R by sin(ry) r² + y2" (x, y) E R? \ {(0,0)}, g(x, y) g(0,0) = 0. %3D Prove that g is separately continuous, but not continuous, at (0,0).

Advanced Engineering Mathematics
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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1. (a) Define f: R² → R by sin(r-y) f(x, y):= if y #x, f(t, t) = 1 Vt ER. Prove that f is continuous at all points of R?. (b) Define g: R2 → R by sin(ry) r² + y2" (x, y) E R? \ {(0,0)}, g(x, y) g(0,0) = 0. %3D Prove that g is separately continuous, but not continuous, at (0,0).
Transcribed Image Text:1. (a) Define f: R² → R by sin(r-y) f(x, y):= if y #x, f(t, t) = 1 Vt ER. Prove that f is continuous at all points of R?. (b) Define g: R2 → R by sin(ry) r² + y2" (x, y) E R? \ {(0,0)}, g(x, y) g(0,0) = 0. %3D Prove that g is separately continuous, but not continuous, at (0,0).
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