Let f : (a, b) → R be continuously differentiable, where −∞ ≤ a < b ≤ +∞. Consider x0 ∈ (a, b) Show that f is locally Lipschitz around x0 i.e. there is a δ > 0 and a C > 0 such that |f(x) − f(y)| ≤ C|x − y|, ∀x, y ∈ [x0 − δ, x0 + δ] Does this imply that f is Lipschitz continuous throughout (a, b)? If no, provide a counterexample.
Let f : (a, b) → R be continuously differentiable, where −∞ ≤ a < b ≤ +∞. Consider x0 ∈ (a, b) Show that f is locally Lipschitz around x0 i.e. there is a δ > 0 and a C > 0 such that |f(x) − f(y)| ≤ C|x − y|, ∀x, y ∈ [x0 − δ, x0 + δ] Does this imply that f is Lipschitz continuous throughout (a, b)? If no, provide a counterexample.
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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Let f : (a, b) → R be continuously differentiable, where −∞ ≤ a < b ≤ +∞. Consider x0 ∈ (a, b) Show that f is locally Lipschitz around x0 i.e. there is a δ > 0 and a C > 0 such that |f(x) − f(y)| ≤ C|x − y|, ∀x, y ∈ [x0 − δ, x0 + δ] Does this imply that f is Lipschitz continuous throughout (a, b)? If no, provide a counterexample.
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