Question:

<div class="solution"><p><strong>Key Idea:</strong> Analyse <span class="math-inline">$\tan^2 x$</span> near <span class="math-inline">$x=0$</span>. As <span class="math-inline">$x\to 0$</span>, <span class="math-inline">$\tan^2 x\to 0^+$</span>, so <span class="math-inline">$[\tan^2 x]=0$</span> near <span class="math-inline">$x=0$</span>.</p><p><strong>Step 1:</strong> <span class="math-inline">$\lim_{x\to 0}[\tan^2 x]=0=f(0)$</span>, so <strong>f is continuous at x=0</strong>. Option (A) wrong, (B) correct.</p><p><strong>Step 2:</strong> For differentiability at <span class="math-inline">$x=0$</span>: since <span class="math-inline">$f(x)=0$</span> in a neighbourhood of 0, <span class="math-inline">$f'(0)=0\neq 1$</span>. Options (C) and (D) wrong.</p><p><strong>Answer: (B)</strong></p></div> <div class="key-concept"><strong>Key Concept:</strong> GIF of a function: behaviour near integers determines continuity</div> <div class="trap-box"><strong>Trap:</strong> The limit of the GIF equals GIF of the limit only when the inner expression doesn't hit an integer in the limit process. Here <span class="math-inline">$\tan^2 x\to 0^+$</span>, not 0⁻, so the floor is 0.</div>