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This is an old revision of this page, as edited by 133.11.80.84 (talk) at 07:05, 3 September 2007 (→Sufficient for continuity?). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Sufficient for continuity?

Regarding "...a function f : X → Y is continuous...if the preimage of every closed set of Y is closed in X." at the end of the second paragraph:

Forgive me if I'm mistaken, but in general, preimages of closed sets being closed does not ensure continuity. E.g. for f: [0,1] -> (0,1] where f(x) = x (for x != 0) and f(0) = 1/2, f is not continuous. Perhaps something is assumed that I missed? 203.150.100.189 08:38, 20 March 2007 (UTC)[reply]

Please read again: "if the preimage of every open set of Y is open in X". For your example, the preimage of ]1/4,3/4[ is then {0}U]1/4,3/4[ which is not open.--133.11.80.84 07:05, 3 September 2007 (UTC)[reply]

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