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Let $X \subset \mathbb{R}$ and let $f,g : X \rightarrow X$ be a continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)= X$. Which one of the following sets cannot be equal to $X$?

1. $[0, 1]$
2. $(0, 1)$
3. $[0, 1)$
4. $\mathbb{R}$

I think answer is D since its given that X is a proper subset of R, then how can it be equal to R.
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but any logic for other options not being ans ?

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