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By Terence Tao

ISBN-10: 8185931631

ISBN-13: 9788185931630

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Extra info for Analysis II (Texts and Readings in Mathematics, No. 38) (Volume 2)

Example text

Show that for each x E R, the function y ~---+ f(x, y) is continuous on R, and for each y E R, the function x ~---+ f (x, y) is continuous on R. Thus a function f(x, y) which is jointly continuous in (x, y) is also continuous in each variable x, y separately. 3. 11. Let f : R 2 -+ R be the function defined by f(x, y) := (x,y) =f. (0,0), and f(x,y) = 0 otherwise. Show that for fixed x E R, the function y t-t f(x, y) is continuous on R, and that ~when ;Jh for each fixed y E R, the function x t-t f (x, y) is continuous on R, but that the function f : R 2 -+ R is not continuous on R 2 .

Show that g o f : X --+ Z is also uniformly continuous. 5. Let (X, dx) be a metric space, and let f : X --+ R and g : X --+ R be uniformly continuous functions. Show that the direct sum fEB g: X--+ R 2 defined by fEB g(x) := (f(x),g(x)) is uniformly continuous. 6. Show that the addition function (x,y) ~--+ x+y and the subtraction function (x, y) ~--+ x- y are uniformly continuous from R 2 toR, but the multiplication function (x, y) ~--+ xy is not. Conclude that if f : X --+ R and g : X --+ R are uniformly continuous functions on a metric space (X, d), then f + g : X --+Rand f- g : X --+ Rare also uniformly continuous.

Then the following four statements are equivalent: (a) f is continuous. (b) Whenever (x(n))~=l is a sequence in X which converges to some point x 0 E X with respect to the metric dx, the sequence (f(x(n)))~=l converges to f(xo) with respect to the metric dy. (c) Whenever V is an open set in Y, the set f- 1(V) := {x E X: f(x) E V} is an open set in X. (d) Whenever F is a closed set in Y, the set X: f(x) E F} is a closed set in X. Proof. 2. f- 1 (F) := { x E 0 422 13. 6. It may seem strange that continuity ensures that the inverse image of an open set is open.

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Analysis II (Texts and Readings in Mathematics, No. 38) (Volume 2) by Terence Tao


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