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zero of a function∗ mathcam† 2013-03-21 16:45:42 closed set Suppose X is a set and f a complex-valued function f : X → C. Then a zero of f is an element x ∈ X such that f (x) = 0. It is also said that f vanishes at x. The zero set of f is the set Z(f ) := {x ∈ X | f (x) = 0}. Remark. When X is a “simple” space, such as R or C a zero is also called a root. However, in pure mathematics and especially if Z(f ) is infinite, it seems to be customary to talk of zeroes and the zero set instead of roots. Examples • For any z ∈ C, define ẑ : X → C by ẑ(x) = z. Then Z(0̂) = X and Z(ẑ) = ∅ if z 6= 0. • Suppose p is a polynomial p : C → C of degree n ≥ 1. Then p has at most n zeroes. That is, |Z(p)| ≤ n. • If f and g are functions f : X → C and g : X → C, then Z(f g) = Z(f ) ∪ Z(g), Z(f g) ⊇ Z(f ), where f g is the function x 7→ f (x)g(x). • For any f : X → R, then Z(f ) = Z(|f |) = Z(f n ), where f n is the defined f n (x) = (f (x))n . ∗ hZeroOfAFunctioni created: h2013-03-21i by: hmathcami version: h34921i Privacy setting: h1i hDefinitioni h26E99i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 • If f and g are both real-valued functions, then Z(f ) ∩ Z(g) = Z(f 2 + g 2 ) = Z(|f | + |g|). • If X is a topological space and f : X → C is a function, then the support of f is given by: supp f = Z(f ){ Further, if f is continuous, then Z(f ) is closed in X (assuming that C is given the usual topology of the complex plane where {0} is a closed set). 2