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Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. || Prove that f' is continuous on R for all v E R² \ {(0,0)}, but that f itself is discon- tinuous at (0, 0) (relative to R²).

Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. || Prove that f' is continuous on R for all v E R² \ {(0,0)}, but that f itself is discon- tinuous at (0, 0) (relative to R²).

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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Please solve (b)

Given a function f : Rm → R" which is continuous on R™ and v E R™ \ {0}. (a) Prove that the function f" : R → R" defined by f" (t) = f (tv) is continuous on R. (b) Define f : R2 → R by letting (y² – x)² (y4 + x2) f(x, y) = if (x, y) # (0,0), and f(0,0) = 1. Prove that fv is continuous on R for all v E R² \ {(0,0)}, but that ƒ itself is discon- tinuous at (0, 0) (relative to R²).
Transcribed Image Text:Given a function f : Rm → R" which is continuous on R™ and v E R™ \ {0}. (a) Prove that the function f" : R → R" defined by f" (t) = f (tv) is continuous on R. (b) Define f : R2 → R by letting (y² – x)² (y4 + x2) f(x, y) = if (x, y) # (0,0), and f(0,0) = 1. Prove that fv is continuous on R for all v E R² \ {(0,0)}, but that ƒ itself is discon- tinuous at (0, 0) (relative to R²).
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