Discussion
Let \(X\) be a connected topological space such that there exists a non-constant continuous function \(f : X \to \mathbb{R}\), where \(\mathbb{R}\) is equipped with the usual topology. Let \(f(X) = \{f(x) : x \in X\}\). Then
Answer: D
Both \(f(X)\) and \(X\) are uncountable.
A continuous image of a connected space is connected, and the connected subsets of \(\mathbb{R}\) are intervals. Since \(f\) is not constant, \(f(X)\) contains two different values and hence the whole interval between them, which is uncountable.
\(f\) maps \(X\) onto \(f(X)\), so \(X\) has at least as many points as \(f(X)\) and is uncountable too.
This rules out every option in which \(f(X)\) or \(X\) is countable.
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