Limits, Continuity & Differentiability
Differentiability from Additive Functional Equations
Grade 12
Question:
<p>Let <i>f</i> : ℝ → ℝ be a function such that <i>f</i>(<i>x</i> + <i>y</i>) = <i>f</i>(<i>x</i>) + <i>f</i>(<i>y</i>), ∀<i>x</i>, <i>y</i> ∈ ℝ. If <i>f</i>(<i>x</i>) is differentiable at <i>x</i> = 0, then which of the following is correct?</p>
<p>(a) <i>f</i>(<i>x</i>) is differentiable only in a finite interval containing zero</p>
<p>(b) <i>f</i>(<i>x</i>) is continuous for all <i>x</i> ∈ ℝ</p>
<p>(c) <i>f</i>'(<i>x</i>) is constant for all <i>x</i> ∈ ℝ</p>
<p>(d) <i>f</i>(<i>x</i>) is differentiable except at finitely many points</p>
Step-by-Step Solution
Key Concept: A function satisfying Cauchy's functional equation f(x+y) = f(x) + f(y) that is differentiable at even one point must be linear f(x) = cx, which implies it is continuous and differentiable everywhere. The key is showing that differentiability at x = 0 forces differentiability at all points.
<p><strong>Step 1: Use the functional equation property</strong></p><p>Given: f(x + y) = f(x) + f(y) for all x, y ∈ ℝ</p><p>Setting y = 0: f(x) = f(x) + f(0), which gives f(0) = 0</p><p><strong>Step 2: Establish differentiability at x = 0</strong></p><p>Since f is differentiable at x = 0:</p><p>f'(0) = lim(h→0) [f(h) - f(0)]/h = lim(h→0) f(h)/h = c (some constant)</p><p><strong>Step 3: Prove f is differentiable everywhere</strong></p><p>For any point x ∈ ℝ:</p><p>f'(x) = lim(h→0) [f(x+h) - f(x)]/h</p><p>Using the functional equation: f(x+h) - f(x) = f(h)</p><p>Therefore: f'(x) = lim(h→0) f(h)/h = f'(0) = c</p><p>This shows f is differentiable at every point with the same derivative.</p><p><strong>Step 4: Prove f(x) = cx</strong></p><p>Since f'(x) = c for all x, integrating: f(x) = cx + k</p><p>Using f(0) = 0: k = 0, so f(x) = cx</p><p><strong>Step 5: Verify continuity</strong></p><p>Since f(x) = cx is a linear function, it is continuous everywhere on ℝ.</p><p><strong>Step 6: Check each option</strong></p><p><strong>Option A:</strong> FALSE - f is differentiable on all of ℝ, not just a finite interval</p><p><strong>Option B:</strong> TRUE - f(x) = cx is continuous for all x ∈ ℝ</p><p><strong>Option C:</strong> TRUE - f'(x) = c is constant for all x ∈ ℝ</p><p><strong>Option D:</strong> FALSE - f is differentiable everywhere, not just except at finitely many points</p><p><strong>∴ Answer: B, C</strong></p>
Correct Answer: B, C