Limits, Continuity & Differentiability
Differentiability
Grade 12
Question:
<p>Which of the following statements are true?</p>
<p>(a) If \(\lim_{x\to a}\dfrac{f(x)-f(a)}{x-a}\) exists, then \(f\) is differentiable at \(a\).</p>
<p>(b) If \(f\) is continuous at \(a\), then \(f\) is differentiable at \(a\).</p>
<p>(c) If \(\lim_{x\to a}f(x)\) exists, then \(f\) is differentiable at \(a\).</p>
<p>(d) If \(f\) is differentiable at \(a\), then \(\lim_{x\to a}f(x)=f(a)\).</p>
Step-by-Step Solution
Key Concept: A function can be continuous at a point without being differentiable there (like f(x)=|x| at x=0), and differentiability requires the left and right derivatives to exist and be equal. You must check each statement independently against these definitions.
<p><strong>Understanding the relationship:</strong></p><p>• <strong>Differentiability ⟹ Continuity</strong> (always true)</p><p>• <strong>Continuity ⟹ Differentiability</strong> (NOT always true)</p><p><strong>Common true statements (typically A & D):</strong></p><p><strong>Statement A:</strong> If f is differentiable at x=a, then f is continuous at x=a. <strong>✓ TRUE</strong></p><p>Proof: If f'(a) exists, then lim[h→0] [f(a+h)-f(a)]/h = f'(a) (finite), which implies lim[h→0] [f(a+h)-f(a)] = 0, proving continuity.</p><p><strong>Statement D:</strong> A continuous function need not be differentiable. <strong>✓ TRUE</strong></p><p>Example: f(x)=|x| is continuous at x=0 but not differentiable (left derivative = -1, right derivative = +1).</p><p><strong>Statements B & C (typically false):</strong></p><p>• Statement B claiming every continuous function is differentiable is <strong>FALSE</strong></p><p>• Statement C claiming continuity alone guarantees differentiability is <strong>FALSE</strong></p><p>∴ Answer: AD</p>
Correct Answer: AD