<p>If \(f(x)\) be a continuous function in \([1, 3]\) and \(f(x)\) takes rational values for all \(x\) in the interval, and \(f(2) = 10\) then \(f(x) = 10\) for all \(x \in [1, 3]\).</p><p><em>State whether this statement is true or false.</em></p>
Step-by-Step Solution
Key Concept: A continuous function on a closed interval that only takes rational values must be constant, since the rationals are disconnected (totally disconnected) in ℝ, and a continuous image of a connected space must be connected.
<p><strong>Step 1:</strong> Recognize that [1,3] is a connected set in ℝ.</p><p><strong>Step 2:</strong> The continuous image of a connected set must be connected. Therefore f([1,3]) is a connected subset of ℝ.</p><p><strong>Step 3:</strong> We're told f(x) ∈ ℚ for all x ∈ [1,3], so f([1,3]) ⊆ ℚ.</p><p><strong>Step 4:</strong> The only connected subsets of ℚ are singleton sets (ℚ is totally disconnected). Therefore f([1,3]) = {c} for some constant c ∈ ℚ.</p><p><strong>Step 5:</strong> Since f(2) = 10 and f([1,3]) is a singleton, f(x) = 10 for all x ∈ [1,3].</p><p>∴ The statement is <strong>TRUE</strong>.</p>
Correct Answer: A