Limits, Continuity & Differentiability
Continuity at a point
Grade 12
Question:
<p>Let <span>\(f:[−1,3]\to\mathbb{R}\)</span> be defined as <span>\(f(x)=\begin{cases}|x|+[x], & -1\leq x<1\\x+|x|, & 1\leq x<2\\x+[x], & 2\leq x\leq 3\end{cases}\)</span> where <span>\([t]\)</span> denotes the greatest integer less than or equal to <span>\(t\)</span>. Then <span>\(f\)</span> is discontinuous at</p>
<p>(a) four or more points</p>
<p>(b) only two points</p>
<p>(c) only three points</p>
<p>(d) only one point</p>
Step-by-Step Solution
Key Concept: The greatest integer function is discontinuous at integer values. Check limits from left and right at each integer point in the domain, including points where the piece-wise definition changes.
<p><strong>Given function:</strong> <span>$f:[−1,3]\to\mathbb{R}$</span> defined as <span>$f(x)=\begin{cases}|x|+[x], & -1\leq x<1\\x+|x|, & 1\leq x<2\\x+[x], & 2\leq x\leq 3\end{cases}$</span></p><p><strong>Step 1:</strong> Rewrite using the definition of greatest integer function. For <span>$n\leq x<n+1$</span> where <span>$n$</span> is an integer, <span>$[x]=n$</span>:</p><p><span>$f(x)=\begin{cases}-x-1, & -1\leq x<0\\x, & 0\leq x<1\\2x, & 1\leq x<2\\x+2, & 2\leq x<3\\6, & x=3\end{cases}$</span></p><p><strong>Step 2:</strong> Check continuity at critical points.</p><p>At <span>$x=0$</span>: <span>$\lim_{x\to 0^-}f(x)=-1\neq f(0)=0$</span>. Discontinuous.</p><p>At <span>$x=1$</span>: <span>$\lim_{x\to 1^-}f(x)=1\neq f(1)=2$</span>. Discontinuous.</p><p>At <span>$x=2$</span>: <span>$\lim_{x\to 2^-}f(x)=4=f(2)=\lim_{x\to 2^+}f(x)=4$</span>. Continuous.</p><p>At <span>$x=3$</span>: <span>$\lim_{x\to 3^-}f(x)=5\neq f(3)=6$</span>. Discontinuous.</p><p><strong>∴ Answer is (c) only three points.</strong></p>
Correct Answer: C