if we know that f : [a,b] ----> reals is continuous. How can we prove that the image f([a,b]) is a closed interval say [m,M] for some m,M in reals?
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A continuous function maps a compact set to a compact set.
Originally Posted by maximus101 if we know that f : [a,b] ----> reals is continuous. How can we prove that the image f([a,b]) is a closed interval say [m,M] for some m,M in reals? Or, perhaps a little simpler answer than HallsofIvy's, put $\displaystyle f(a)=m\,,\,f(b)=M$ and now use the intermediate value theorem for continuous functions... Tonio
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