Limits, Continuity & Differentiability
Differentiability at Points
Grade 12
<p><strong>Example 37:</strong> Match the column I with column II.</p><p><strong>Column I:</strong></p><p>(i) \(\sin(\pi[x])\)</p><p>(ii) \(\sin\{(x-[x])\pi\}\)</p><p><strong>Column II:</strong></p><p>(A) differentiable everywhere</p><p>(B) nowhere differentiable</p><p>(C) not differentiable at \(-1\) and \(+1\)</p><p>where \([\cdot]\) denotes greatest integral function.</p>
Step-by-Step Solution
Key Concept: Recognize that $[x]$ produces integers only, making $\sin(\pi[x])$ constant. The fractional part $\{x\}$ has jump discontinuities at integers.
<p><strong>Solution for (i):</strong></p><p>We know $[x] \in \mathbb{I}$ for all $x \in \mathbb{R}$.</p><p>Therefore, $\sin(\pi[x]) = \sin(n\pi) = 0$ for all $x \in \mathbb{R}$ (where $n$ is an integer).</p><p>By theory, every constant function is differentiable in its domain.</p><p>Hence, $\sin(\pi[x])$ is <strong>differentiable everywhere</strong>.</p><p>Thus, (i) → (A)</p><p><strong>Solution for (ii):</strong></p><p>We know $x - [x] = \{x\}$ (fractional part of $x$).</p><p>Then $\pi(x - [x]) = \pi\{x\}$.</p><p>The fractional part $\{x\}$ is not differentiable at integral points ($x \in \mathbb{Z}$).</p><p>Therefore, $f(x) = \sin\{\pi(x - [x])\}$ is not differentiable at $x \in \mathbb{Z}$.</p><p>Thus, (ii) → (C)</p>
Correct Answer: (i) → (A), (ii) → (C)