TIFR-2015-Maths-B-12

Question

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 $X$?

• A. $[0, 1]$
• B. $(0, 1)$
• C. $[0, 1)$
• D. $\mathbb{R}$

Answer 1

I think answer is D since its given that X is a proper subset of R, then how can it be equal to R.

Comments

but any logic for other options not being ans ?