 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2009/e/e4/a/ii/solution

Suppose

f (X) = U \cup V

were a separation into two disjoint nonempty relatively open sets. Continuity makes

f^{- 1} (U)

and

f^{- 1} (V)

open in

X

, and they are disjoint, nonempty and cover

X

. This contradicts connectedness of

X

. Hence

f (X) is connected

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