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Calculus Continuity question?
If f is continuous and 0 <= f(x) <= 1 for x belongs to [0, 1], then prove that there exists c in [0, 1] such that f(c) = c.
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- Rich JLv 61 decade agoFavorite Answer
consider g(x) = f(x) - x for x in [0,1], and g is continuous on this interval.
we also have -1 =< g(x) =< 1.
by the intermediate value theorem, there is a value c in [0,1] such that g(c)=0.
so 0 = g(c) = f(c) - c, ie f(c) = c.
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