Limits, Continuity & Differentiability
Continuity and Differentiability of Piecewise Function
nta_pyq_2025_apr
Grade 12
Question:
Let $f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1+x+[x],\, x+2[x]\}, & 0 \leq x < 2 \\ 5, & x > 2 \end{cases}$ where $[\cdot]$ denotes the greatest integer function. If $\alpha$ and $\beta$ are the number of points where $f$ is not continuous and not differentiable, respectively, then $\alpha + \beta$ equals ___
Step-by-Step Solution
Key Concept: Simplify $f$ on $[0,2)$ by checking $[x]=0$ for $x\in[0,1)$ and $[x]=1$ for $x\in[1,2)$. Identify discontinuities and non-differentiable points.
$f(x)=\begin{cases}3x & x<0\\ x & 0\leq x<1\\ x+2 & 1\leq x<2\\ 5 & x>2\end{cases}$. Discontinuous at $x=1$ (jump from 1 to 3) and $x=2$ (approaches 4, value undefined). $\alpha=2$. Non-differentiable at $x=0$ (slope change), $x=1$, $x=2$. $\beta=3$. $\alpha+\beta=5$.
Correct Answer: 5