推 dearcherry11:感謝Z大!! 03/12 22:38
Denote f([a,b]) = Imf
Since [a,b] is compact, f is continuous
by extreme value theorem, there exists c,d€[a,b]
s.t. f(c)=<f(x)<=f(d) , for all x€[a,b] ---(*)
so Imf⊆[f(c),f(d)]
Since [a,b] is connected, f is continuous
we have Imf is connected, too.
Since from (*) we know f(c),f(d)€Imf
so [f(c),f(d)] ⊆ Imf (if not, there exists two open sets seperate Imf)
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