Regarding existence of continuous function which is onto

Question

I am solving previous years problems in analysis/ topology of my institute and I am doubtful about this answer of this question. Question is - Can there exists a continuous function from [-1,1] to R ( Set of Real numbers) which is onto. I think there can exists as any interval is equivalent ( cardinality wise) to set of real numbers. But the answer is no. Please help.

Answer 1

One can show that the continuous image of a compact set is again compact, see for example https://math.stackexchange.com/a/226328/450140. As $\mathbb{R}$ is not compact with the standard topology this proves that such map can not exist.

Answer 2

A theorem in analysis states that a continuous function on a compact set (e.g. closed interval) is bounded and attains its minimum and maximum. Because $\mathbb{R}$ is unbounded, such a function cannot map to all of $\mathbb{R}$.