1 La Let F R R Show That If Dif X 0 For All X Er Then F Is Independent Of The The I Th Variable Independe 1 (55.85 KiB) Viewed 28 times
1. (la) Let f: R" → R. Show that if Dif(x) = 0 for all x ER", then f is independent of the the i-th variable. (Independent of the i-th variable means f(x₁, ..., X;,...,xn) = ƒ (X1₁,..., X₁-1, y, Xi+1...,xn) for all x = (x₁,...,x) = R¹ and y € R.) (1b) Let f: R" →→ R. Show that if Dif(x)=0 for all x € R" and for all i = 1,..., n, then f is constant.
(1c) Is part (b) true if the domain of f is not all of R"? (1d) Is part (a) true if the domain of f is not all of R" but still open and path connected? Hint: Find a counterexample with the domain S = R² \ {(x, y) = R² : x ≥ 0, y = 0}
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