Continuity and Differentiability
Differentiability of Fractional Part Functions
JEE Main
Grade 12
Question:
The number of points at which $f(x) = \max\{x - [x], -x + [x]\}$, where $[x]$ denotes the floor function, is not differentiable in $(0, 3)$ is _____.
Step-by-Step Solution
Key Concept: $f(x) = \max\{\{x\}, 1 - \{x\}\}$ where $\{x\} = x - [x]$. On each interval $[n, n + 1)$: $f(x) = \max(x - n, 1 - (x - n))$. This has a corner at the midpoint $x = n + \frac{1}{2}$ and a jump discontinuity at each integer $n > 0$. Non-differentiable points in $(0, 3)$: $\{1/2, 1, 3/2, 2, 5/2\}$ — that is 5 points.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: 5