DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION
... that xn → x when n → ∞; then f (xn ) = 0 for every n and the sequence {f (xn )} does not converge to f (x) = 1/q, so f is not continuous at x. On the other hand, for x ∈ R \ Q, let us see that f is continuous at x by checking that f (xn ) → f (x) = 0 for every sequence {xn } that tends to x. As f (y ...
... that xn → x when n → ∞; then f (xn ) = 0 for every n and the sequence {f (xn )} does not converge to f (x) = 1/q, so f is not continuous at x. On the other hand, for x ∈ R \ Q, let us see that f is continuous at x by checking that f (xn ) → f (x) = 0 for every sequence {xn } that tends to x. As f (y ...
Proofs - Maths TCD
... Proof. Suppose X is Hausdorff and A ⊂ X is compact. To show that X − A is open, let x ∈ X − A be given. Then for each y ∈ A there exist disjoint open sets Uy , Vy such that x ∈ Uy and y ∈ Vy . Since the sets Vy form an open cover of A, finitely many of them cover A by compactness. Suppose that Vy1 , ...
... Proof. Suppose X is Hausdorff and A ⊂ X is compact. To show that X − A is open, let x ∈ X − A be given. Then for each y ∈ A there exist disjoint open sets Uy , Vy such that x ∈ Uy and y ∈ Vy . Since the sets Vy form an open cover of A, finitely many of them cover A by compactness. Suppose that Vy1 , ...
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... The ordered pairs of the function f are reversed to produce the ordered pairs of the inverse relation. Example: Given the function f = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4} and its range is {1, 2, 3}. The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}. The domain o ...
... The ordered pairs of the function f are reversed to produce the ordered pairs of the inverse relation. Example: Given the function f = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4} and its range is {1, 2, 3}. The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}. The domain o ...
G.1 Normality of quotient spaces For a quotient space, the
... topology coherent with these subspaces; it is called an (infinite-dimensional) ...
... topology coherent with these subspaces; it is called an (infinite-dimensional) ...
