Problem 4 Let D Crm Be Open And Let F D R Be Such That All Partial Derivatives Of All Components Of F Exist And A 1 (103.5 KiB) Viewed 25 times
• Problem 4. Let D CRM be open and let f → R" be such that all partial derivatives of all components of f exist and are continuous on D. Let pe D be arbitrary. (1) Show that there exists some r >0 so that Bp() CD. (2) Show that the closed ball B (r/2) is a subset of D. (3) Explain why all partial derivatives of all components of f are bounded on the closed ball B, (r/2). (Hint: What do we know about continu- ous functions on closed and bounded sets of finite dimensional vector spaces?) (4) Explain why all partial derivatives of all components of f are bounded on the open ball Bp(r/2). (5) Explain why there exists some constant K > 0 (dependent on p) so that for all u, v E Bp(r/2) we have ||| (u) - f(0)|| <K || 2 – v ||. (Hint: apply a result from the lectures/course notes regarding functions that have bounded partial derivatives of all their components).
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→ R" be such that all partial derivatives of all components of f exist and are continuous on D. Let pe D be arbitrary. (1) Show that there exists some r >0 so that Bp() CD. (2) Show that the closed ball B (r/2) is a subset of D. (3) Explain why all partial derivatives of all components of f are bounded on the closed ball B, (r/2). (Hint: What do we know about continu- ous functions on closed and bounded sets of finite dimensional vector spaces?) (4) Explain why all partial derivatives of all components of f are bounded on the open ball Bp(r/2). (5) Explain why there exists some constant K > 0 (dependent on p) so that for all u, v E Bp(r/2) we have ||| (u) - f(0)|| <K || 2 – v ||. (Hint: apply a result from the lectures/course notes regarding functions that have bounded partial derivatives of all their components).