Continuous Functions

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Originally Posted by acevipa

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I'll show you number 1.

Define the function $g(a)=f(a)-f(a+1/2)$ and note it is continuous. Then, we have $g(0) = f(0)-f(1/2)$ and $g(1/2) = f(1/2)-f(1)$ . Depending on the size of f(1/2) relative to f(0)=f(1), we either have g(a) goes from negative to positive or positive to negative. In either case, we have that g(a) = 0 at some point from 0 to 1/2. This shows number 1.