"Prove that if f,g are continuous functions on [0,1] with range [0,1], then there exists a point c such the "
This is a variation on a popular problem given to encourage students to use the [[wiki|intermediate value theorem|intermediate value theorem]]. The proof involves a function , this function can't be exclusively positive or negative, so we can find c so that .
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The proof is really about getting the given situation in to a form where it will "plug in" to the definition nicely. It has no deeper insight about why this happens. For that we must look at the fixed point theorems.
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