Discussion
Consider the following statements :
(a) There is a continuous map from [0, 1] onto (0, 1)
(b) There is a continuous map from (0, 1) onto [0, 1]
Then :
Answer: B
The answer is only (b) is true.
(a) is false: [0, 1] is compact and the continuous image of a compact set is compact, but (0, 1) is not compact. (Equivalently, a continuous function on [0, 1] attains its maximum and minimum, so its image is a closed interval.)
(b) is true: for example \(f(x) = \tfrac{1}{2}(1 + \sin(4\pi x))\) is continuous on (0, 1) and takes the values 0 and 1 inside the interval, at \(x = 3/8\) and \(x = 1/8\), so its image is all of [0, 1]. Compactness is not needed in the domain to reach a compact image.
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