Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>Which statements are correct?<br>(A) ∃ f:[0,1]→ℝ discontinuous everywhere with |f| continuous everywhere<br>(B) F=f·g, f diff at x=a, f(a)=0, g continuous at x=a ⟹ F diff at x=a<br>(C) Rf'(a)=2, Lf'(a)=3 ⟹ f non-diff at x=a but always continuous<br>(D) f(a) and f(b) have opposite signs ⟹ ∃ solution of f(x)=0 in (a,b) if f continuous on [a,b]</p>
A only
B,C
A,B,C
<strong>A,B,C,D</strong>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>(A) TRUE: e.g., f(x)=1 if x∈Q, -1 if x∉Q. |f|=1 everywhere continuous, f discontinuous everywhere.</p><p>(B) TRUE: F'(a)=lim[f(x)g(x)-0]/[x-a]=f'(a)·g(a)·... more carefully: F(x)-F(a)=(f(x)-f(a))g(x)+f(a)(g(x)-g(a))=(f(x)-f(a))g(x). Divide by (x-a): →f'(a)·g(a). Exists since g continuous. ✓</p><p>(C) TRUE: If Rf'≠Lf', f is non-differentiable but continuous at x=a (existence of both one-sided derivatives implies continuity). ✓</p><p>(D) TRUE: Intermediate Value Theorem. ✓</p><p><strong>Answer: (A),(B),(C),(D)</strong></p><div class="key-concept"><strong>Key Concept:</strong> Existence of both one-sided derivatives ⟹ continuity at that point</div></div>
Correct Answer: A,B,C,D