The sketch of $y=\tan^{-1} \left ( \frac{1}{x} \right )$ For this function, the horizontal asymptotes are easy enough to find. However, I have issue with the point $x=0$.
Graphing calculators say that there is a vertical line connecting $y=\frac{\pi}{2}$ down to $y=-\frac{\pi}{2}$. However, this will imply that for each $y\in \left [0,\frac{\pi}{2} \right )$, there will be two $x$ values satisfying this, which doesn't seem right to me because if I set $y=1$, there is only one solution, which is $x=\frac{\pi}{4}$.
What is going on here? I am thinking that $x=0$ is undefined, so there is a jump discontinuity.
If what I am thinking is correct, why is it that graphing calculators (Wolframalpha and Graphmatica) point towards having a vertical line there?
 A: WolframAlpha clearly states below the graph that the domain is $\mathbb{R} \setminus \{0\}$. It also states that the range is $(-\pi/2,0) \cup (0,\pi/2) $.
Perhaps the vertical line is to emphasize the jump discontinuity (non-removable). It would be nearly impossible to see any graph that stops short of the y-axis. (I.e. where would we put the points closest to $x = 0$, without there being a a point even closer at $x = 0$, but not yet $0$?)
A: There is no vertical line joining these two "y-intercepts" There is a non removable discontinuity taking place on the y-axis. With the limit definition of the derivative you can find that both tangents at the "y-intercetps" have slope -1 But obviously the derivative as a function does not exist because the given function is and remains discontinuous
A: Graphing calculators are usually unable to deal with discontinuities, they just keep on plotting and connecting points, so you have to know this "feature" and guess there is something going wrong, or you can see it directly by inspecting the function, since
$$\arctan x \underset{x\to+\infty}{\longrightarrow}+\frac{\pi}2$$
$$\arctan x \underset{x\to-\infty}{\longrightarrow}-\frac{\pi}2$$
Some more explanation: to plot a curve, a calculator (and most plotting software, except maybe clever CAS) pick some points in the $x$-range (typically $\sim 100$), compute the $y$ values, and plot the points, connecting them by lines so the rendering is more pretty. It works like a charm with continuous functions, but it fails miserably when there are discontinuities, as there is then an almost vertical line connecting two consecutive points.
For example, it's easy to change from lines to only points in a spreadsheet like Excel or LibreOffice, and see the difference.
Sometimes, plotting software is able to recognize that a big jump is probably a discontinuity, and they don't connect these two consecutive points. But calculators have limited capabilities, and are not likely to have this.
Here is the output from Maxima, with plot2d(atan(1/x),[x,-1,1]);

That even WolframAlpha is not able to see this is rather funny, but it's not the first time WA shows inconsistent behaviour. The key point is: never, ever trust a mathematical answer from software, even CAS with good reputation. They all have bugs, some less than others, but all have some, and you will only know by experience what kind of output you can trust, from a program.
For example, I took Maxima for this plot, but I know it has disastrous problems for managing expressions with square roots, especially integrals.
A: $$x \to 0^\pm \implies \frac{1}{x} \to \pm\infty \implies \tan^{-1}(\frac{1}{x}) \to \pm \frac{\pi}{2}$$
Alternatively, to emphesize the fail of $y$ "at" $0$:
$$ x = \frac{1}{\tan(y)} \to 0 \implies \tan(y) \to \pm \infty \implies y \to \pm \frac{\pi}{2}$$
