Animated Solution for Mathematics - Limits, Continuity and Differentiability: Draw a graph of the function $y=[x]+|1-x|,-1\le x\le 3$. Determine the points, if any, where this function is not differentiable.

The function $y=[x]+|1-x|$ is a piecewise function defined over the domain $[-1,3]$. To analyze its behavior, we must decompose the Greatest Integer Function $[x]$ and the Modulus Function $|1-x|$ across the sub-intervals of the domain.

The Greatest Integer Function $[x]$ creates a staircase effect, jumping at every integer value. The Modulus Function $|1-x|$ acts as a V-shaped graph with its vertex at $x=1$.

We analyze the function segment by segment to observe how the components interact:

For the interval $[-1,0)$: Here, $[x]=-1$. Since $x<1$, the expression $1-x$ is positive, so $|1-x|=1-x$. The function becomes:

For the interval $[0,1)$: Here, $[x]=0$. Since $x<1$, the expression $1-x$ remains positive, so $|1-x|=1-x$. The function becomes:

For the interval $[1,2)$: Here, $[x]=1$. Since $x>1$, the expression $1-x$ is negative, so $|1-x|=-(1-x)=x-1$. The function becomes:

For the interval $[2,3)$: Here, $[x]=2$. Since $x>1$, the expression $|1-x|=x-1$. The function becomes:

A function is differentiable only if it is continuous and smooth. At every integer point, the Greatest Integer Function $[x]$ undergoes a jump discontinuity.

For example, at $x=0$: The left-hand limit is ${\lim }_{x\to 0^{-}}(-x)=0$. The right-hand limit is ${\lim }_{x\to 0^{+}}(1-x)=1$. Since the limits do not match, the function is discontinuous at $x=0$.

Because a jump discontinuity prevents the existence of a tangent line, the function cannot be differentiable at these points. By evaluating the behavior at each integer within the domain, we observe that the function is discontinuous at $x=0,1,2,3$.

Therefore, the points of non-differentiability for the function $y=[x]+|1-x|$ on the interval $[-1,3]$ are $x\in \{0,1,2,3\}$.