20. Assume that U is a connected open subset of R and f: U Rm is...

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20. Assume that U is a connected open subset of R" and f: U Rm is differen tiable everywhere on U. If (Df)p = 0 for all p = U, show that f is constant. 21. For U as above, assume that f is second-differentiable everywhere and (D f)p = 0 for all p. What can you say about f? Generalize to higher-order differentia- bility. 20. Assume that U is a connected open subset of R" and f: U Rm is differen tiable everywhere on U. If (Df)p = 0 for all p = U, show that f is constant. 21. For U as above, assume that f is second-differentiable everywhere and (D f)p = 0 for all p. What can you say about f? Generalize to higher-order differentia- bility.

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20 To show that f is constant when Dfp 0 for all pU we can use the Mean Value Theorem Since U is a c... View the full answer