Show that the continuous image of a disconnected set may be connected. Show that

ID: 3082294 • Letter: S

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Show that the continuous image of a disconnected set may be connected. Show that this can happen even if f is a bijection.

Explanation / Answer

Consider the set f : R-{0}->R So R-{0} is disconnected define f(x) = |x| So range of f is [0, infinity) which is connected. f : R-[-1, 0) -> R a disconnected set Now f is a bijection f(x) = log (1 + x) when x >= 0 .. This covers from [0, infinity) f(x) = -x+1 where x lies in (-infinity, -1) This covers from (-infinity, 0) So this is a bijection and from a disconnected set to a connected set. This is the required answer message me if you have any doubts

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Show that the continuous image of a disconnected set may be connected. Show that

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