draw the graph of a function with all the following properties. type anything in the box and show your graph…

Explanation:

Step1: Understand removable discontinuity

A removable discontinuity at $x = 1$ means the limit exists as $x\to1$ but the function value at $x = 1$ is either different from the limit or undefined.

Step2: Understand non - differentiability at $x = 0$

A function is non - differentiable at a point if there is a sharp corner, cusp, vertical tangent or a discontinuity at that point. We can create a sharp corner at $x = 0$.

Step3: Sketch the function

We can start by drawing a curve that is defined on $[-3,3]$. For example, we can consider a piece - wise function. Let $f(x)=\begin{cases}x + 1, &x\in[-3,0)\2, &x = 0\x - 1, &x\in(0,1)\cup(1,3]\end{cases}$ and define $f(1)$ to be something different from $\lim_{x\to1}f(x)=0$ (say $f(1)=1$) to create the removable discontinuity at $x = 1$. The sharp corner at $x = 0$ makes it non - differentiable at $x = 0$.

Answer:

A possible graph can be created from the piece - wise function $f(x)=\begin{cases}x + 1, &x\in[-3,0)\2, &x = 0\x - 1, &x\in(0,1)\cup(1,3]\end{cases}$ with $f(1)=1$ (the actual function can vary as long as it satisfies the three given conditions).