Question:

<div class="solution"><p><span class="math-inline">\(\text{sgn}(x^5)=\text{sgn}(x)\)</span> (since x⁵ has same sign as x). So f(x)=sgn(x).</p><p>(A) f'(0⁺)=1: sgn(x) is constant 1 for x>0, so f'(0⁺)=0, not 1. FALSE ✓</p><p>(B) f'(0⁻)=-1: similarly f'(0⁻)=0. FALSE ✓</p><p>(C) f is continuous but not differentiable at x=0: f is DISCONTINUOUS at x=0. FALSE ✓</p><p>(D) f is discontinuous at x=0: TRUE. So (D) is not false.</p><p><strong>Answer: (A),(B),(C) are false</strong></p><div class="trap-box"><strong>Trap:</strong> sgn(x⁵)=sgn(x) — the odd power preserves sign.</div><div class="key-concept"><strong>Key Concept:</strong> sgn(xⁿ) for odd n equals sgn(x)</div></div>