Can a graph of a continuous function intersect its vertical asymptote? I know a graph can intersect its horizontal asymptote (1, 2, 3).
I think it's possible for a graph of a function to intersect its vertical asymptote. Example: Define $f:\mathbb{R}\rightarrow \mathbb{R}$ by $f(x)=1/x$ for $x\neq 0$ and $f(x)=0$ for $x=0$. Then $x=0$ is a vertical asymptote for and intersects $f$. (Correct me if this example is mistaken.)
I think though that it's not possible for a graph of a continuous function to intersect its vertical asymptote. Is this true? How do I prove this?

No.
By definition, a function $f$ has a vertical asymptote at $a$ if $$\lim_{x\to a} f(x) = \pm \infty$$
Using this definition, it can easily be shown that, if a function has a vertical asymptote at $a$ and the function is also defined at $a$, then the function is not continuous. So, the function's graph cannot intersect the vertical asymptote because in order for it to do so, the graph should contain a point on the asymptote.
