Chapter 3.2, Problem 11E

To determine

Whether the given function $y$ is a corner, a cusp, a vertical tangent, or a discontinuity.

Answer to Problem 11E

The given function $y$ is a discontinuity.

Explanation of Solution

Given information:

The given function is $y=\{\begin{array}{l}{\tan }^{-1}x,x\ne 0\\ 1,x=0\end{array}$

Calculation:

The given function is:

  $y=\{\begin{array}{l}{\tan }^{-1}x,x\ne 0\\ 1,x=0\end{array}$

Since, it is known that the function $\tan x$ is one to one, continuous and unbounded over the interval $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ .

So, the inverse of the function $\tan x$ is also continuous and one to one.

Thus, the given function $y={\tan }^{-1}x$ is continuous and differentiable for all $x$ .

Also, the given function not differentiable because of a jump discontinuity.

Since, ${\tan }^{-1}\left(0\right)=0$ but $y\left(0\right)=1$ as given in the function.

So, the given function $y$ is a discontinuity as shown in graph.

  

Hence, the given function $y$ is a discontinuity.