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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1e46184-5bb3-4339-852b-bb3ef2e0784b%2F0f759e0b-ee69-4e23-a006-e859dd3ccf8c%2Fqexzik_processed.png&w=3840&q=75)
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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