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• CommentRowNumber1.
• CommentAuthorTobyBartels
• CommentTimeNov 25th 2009

I mentioned the intermediate value theorem at pentagon decagon hexagon identity and then began an article on it.

• CommentRowNumber2.
• CommentAuthorColin Tan
• CommentTimeDec 26th 2013
I notice that Theorem 2 already implies that there some c of the unit interval such that f(c) is strictly between -1 and 1. Although we may not be able to construct all values between -1 and 1 to lie in the image of f, at least we can construct one value.
• CommentRowNumber3.
• CommentAuthorTobyBartels
• CommentTimeDec 28th 2013

Right, and more specifically we can construct a $c$ whose value is as close as we like (within $\epsilon$) of our desired value.