Can an open circle be marked with an asymptote?

Question

If I have a function: $$f(x) = \begin{cases} x + 1, & \text{if $x$ < 0} \\ x - 1, & \text{if $x$ > 0} \end{cases}$$

Creating the graph: Graph image, would it be correct to place an asymptote at $x=0$, or must you use open circles at $(0, 1)$ and $(0, -1)$?

Answer 1

For discontinuities this is my favoured presentation :

• a semi-open circle at the discontinuity $\lim\limits_{x\to 1^-}f(x)=1$
• a filled circle at the continuity point $\lim\limits_{x\to 1^+}f(x)=-2$ and $f(1)=-2$

For the function you gave in your post, this would be two semi-open circles since your function is not defined at $x=0$.

When the function can be extended by continuity but has a hole, i.e. a point where the function is not defined, the use a small not filled circle like above.

$\lim\limits_{x\to 5}f(x)$ exists, but $f(5)$ is undefined.

Finally asymptotes can be marked like the pink dashed lines above when the function either has limits in $\pm\infty$ or takes infinite values at a specific $x_0$ ($x_0=-2$ on the picture).

Answer 2

You must use open circles. Placing a vertical(I guess, you didn't specify) asymptote at $x=0$ would say that the graph of the function approaches infinity as $x$ gets closer to 0, which it does not.