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  4. The function y=[x]+|1-x|,-1 ≤ x ≤ 3 Then f(x) is not differentiable at (where [ ⋅ represents greatest integer function)

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Q. The function $y=[x]+|1-x|,-1 \leq x \leq 3$ Then $f(x)$ is not differentiable at (where [ $\cdot$ represents greatest integer function)

Continuity and Differentiability

Solution:

We have $y=[x]+|1-x|,-1 \leq x \leq 3$
or $y = \begin{cases} -1+1-x, & \text{ $-1 \leq x<\,0$} \\[2ex] 0+1-x, & \text{ $1 \leq x<\,1$ }\\[2ex] -1-1+x, & \text{ $1 \leq x<2$} \\[2ex] 2-1+x, & \text{ $2 \leq x < 3$} \\[2ex] 3- 1+x, & \text{ $x=3$} \\[2ex] \end{cases}$
or $y = \begin{cases} -x, & \text{ $-1 \leq x<\,0$} \\[2ex] 1-x, & \text{ $0 \leq x<\,1$ }\\[2ex] x, & \text{ $1 \leq x<2$} \\[2ex] 1+x, & \text{ $2 \leq x < 2$} \\[2ex] 5, & \text{ $x=3$} \\[2ex] \end{cases}$
Clearly, $f (x)$ is discontinuous at x = 0,1, 2, 3 and hence non-differentiable