Limits, Continuity & Differentiability
Continuity and Connectedness
Grade 12
<p>Let \(f: [1, 10] \to \mathbb{Q}\) be a continuous function and \(f(1) = 10\), then \(f(10)\) is equal to</p>
Step-by-Step Solution
Key Concept: A continuous function from a connected domain [1,10] to ℚ (rationals) must be constant, since the image of a connected set under a continuous function is connected, and the only connected subsets of ℚ are singletons.
<p><strong>Step 1:</strong> Recall that [1, 10] is a connected subset of ℝ (it's an interval).</p><p><strong>Step 2:</strong> If f: [1, 10] → ℚ is continuous, then the image f([1, 10]) must be a connected subset of ℚ (since the continuous image of a connected set is connected).</p><p><strong>Step 3:</strong> Identify which subsets of ℚ are connected. The rationals ℚ with standard topology are totally disconnected—every point is isolated. A connected subset of ℚ can only be a single point (a singleton).</p><p><strong>Step 4:</strong> Since f([1, 10]) is connected and contained in ℚ, we must have f([1, 10]) = {c} for some constant c ∈ ℚ.</p><p><strong>Step 5:</strong> This means f is a constant function: f(x) = c for all x ∈ [1, 10].</p><p><strong>Step 6:</strong> Given that f(1) = 10, we have c = 10.</p><p><strong>Step 7:</strong> Therefore, f(10) = 10.</p><p><strong>∴ Answer:</strong> b</p>
Correct Answer: b