Let , where denotes greatest integer function. If and are the number of points, where f is not continuous and is not differentiable, respectively, then equals __________. [2025]

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Solution

Q.
Let $f (x)$ $= {\begin{matrix} 3 x, & x < 0 \\ \min {1 + x + [x], x + 2 [x]}, & 0 \leq x \leq 2 \\ 5, & x > 2 \end{matrix}$ , where $[\cdot]$ denotes greatest integer function. If $α$ and $β$ are the number of points, where f is not continuous and is not differentiable, respectively, then $α + β$ equals __________. [2025]

Ans.
(5)

$f (x)$ $= {\begin{matrix} 3 x, & x < 0 \\ \min {1 + x + [x], x + 2 [x]}, & 0 \leq x \leq 2 \\ 5, & x > 2 \end{matrix}$

$\Rightarrow$ $f (x)$ $= {\begin{matrix} 3 x, & x < 0 \\ x, & 0 \leq x < 1 \\ x + 2, & 1 \leq x < 2 \\ 5, & x > 2 \end{matrix}$

$∴$ By graph, we have f(x) is not continuous at $x \in {1, 2} \Rightarrow α = 2$

f(x) is not differentiable at $x \in {0, 1, 2} \Rightarrow β = 3$

$∴ α + β = 5$ .

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