<p>The inverse function of a continuous function is continuous.</p><p><em>State whether this statement is true or false.</em></p>
Step-by-Step Solution
Key Concept: A continuous function has a continuous inverse if and only if the function is bijective (one-to-one and onto) from its domain to its range. For functions from ℝ to ℝ, a strictly monotonic continuous function guarantees a continuous inverse.
<p><strong>Step 1:</strong> Recall the theorem: If f: [a,b] → ℝ is continuous and strictly monotonic, then f⁻¹ is continuous on f([a,b]).</p><p><strong>Step 2:</strong> For the statement to be universally true, it must hold for ALL continuous functions. Consider f(x) = {x if x ≤ 0; 1+x if x > 0}. This is continuous on ℝ but not strictly monotonic.</p><p><strong>Step 3:</strong> A continuous function need not be bijective. For example, f(x) = sin(x) is continuous but not one-to-one on ℝ, so f⁻¹ as a function doesn't exist on all of ℝ.</p><p><strong>Step 4:</strong> The correct statement is: <strong>If f is continuous and strictly monotonic</strong> (or bijective with additional structure), then f⁻¹ is continuous. But continuity alone is NOT sufficient.</p><p>∴ Answer: <strong>FALSE</strong> - The statement as given is too broad and incorrect without the monotonicity or bijectivity condition.</p>
Correct Answer: A