Given a function f : Rm → R" which is continuous on Rm and v E Rm \ {0}. Prove that the function fº : R → R" defined by fº(t) = f(tv) is continuous on R. Define f : R2 → R by letting (a) (b) (y? – a)? (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).
Given a function f : Rm → R" which is continuous on Rm and v E Rm \ {0}. Prove that the function fº : R → R" defined by fº(t) = f(tv) is continuous on R. Define f : R2 → R by letting (a) (b) (y? – a)? (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
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f^v (t) is just f(tv) where f(x) is a continuous function

Transcribed Image Text:Given a function f : Rm → R" which is continuous on R™ and v E R™m \ {0}.
(a)
Prove that the function f" : R → R" defined by f"(t) = f(tv) is continuous on R.
(b)
%3D
Define f : R? –→ R by letting
(y² – x)²
f (x, y) =
(y4 + x²)
|
if (x, y) # (0, 0), and f(0,0) = 1.
Prove that fº 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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