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E E E
f −1 ([1, +∞[) ⊆ E (¯x, ¯y) ∈ f −1 ([1, +∞[)
{(xn, yn)}n∈N E (xn, yn) → (¯x, ¯y) f(¯x, ¯y) ≥ 1
¯x 2 + cos(¯y) ≥ 1
¯x ≥ 0 xn = ¯x + 1/n yn = ¯y (xn, yn) → (¯x, ¯y) x 2 n > ¯x 2
f(xn, yn) > f(¯x, ¯y) ≥ 1 f(xn, yn) > 1 {(xn, yn)}n∈N ⊆ E
E (¯x, ¯y) (¯x, ¯y) ∈ E
¯x < 0 xn = ¯x − 1/n yn = ¯y (xn, yn) → (¯x, ¯y) x 2 n > ¯x 2
f(xn, yn) > f(¯x, ¯y) ≥ 1
E = f −1 ([1, +∞[) = {(x, y) ∈ R 2 : x 2 + cos(y) ≥ 1}.
E
E
R 2 \ E = R 2 \ f −1 (]1, +∞[) = f −1 (] − ∞, 1]),
E E
∂E = E ∩ R 2 \ E = E ∩ (R 2 \ E) = f −1 (] − ∞, 1]) ∩ f −1 ([1, +∞[) = f −1 (1),
∂E = {(x, y) ∈ R 2 : f(x, y) = 1}
(X, d) {xn}n∈N X x ∈ X
E = {xn : n ∈ N} ∪ {x}
E x
(X, d) E ⊆ X Ē = {x ∈ X : inf{d(x, y) : y ∈
E} = 0}
dE(x) = inf{d(x, y) : y ∈ E} x ∈ Ē dE(x) >
0 x X \ E x /∈ Ē
dE(x) = 0 x /∈ Ē x X \E
δ > 0 x E dE(x) ≥ δ > 0
dE(x) = 0