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I have to find all continuous functions $f : [0,1] \to \mathbb Q$ such that $f(\frac{1}{2})=\frac{2015}{2016}$. but I'm stuck and I don't even know how to start. can someone explain me in details how to do this?

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Use the intermediate value property. There always exists an irrational between two rationals. Use this fact to conclude that such an $f :[0,1]\to \mathbb{Q}$ must be constant. since $f(1/2)=2015/2016$ so $f(x)=2015/2016$ for all $x\in [0,1]$.

Hence there is only one such function.

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The image of $f$ must be connected. But the only non-empty connected subsets of $\Bbb Q$ are singletons.

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