Graph of a function - asymptotes properties

Question

Is it possible that the graph of a function has vertical asymptote if $D_{f} = \mathbb{R}$?

Also, is it possible that the graph of a function intersects its asymptote? (horizontal, slant or vertical)

The answer to both questions seems a straight-forward NO to me, but can someone help prove that? Thanks!

Answer 1

If a function is continuous on $\mathbb{R},$ then doesn't have a vertical asymptote. If $f$ is not continuous, it can happen what explained in comments.

It is not possible that a graph of a function cuts its vertical asymptote. That would contradict the definition of the function.

For the second part, a quote from Wikipedia:

Asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.

The distance between the courbe and the asymptote (horizontal or oblique/slant) has to tend to $0$ with $x\to\infty,$ intersections are allowed.
As an example, consider $$f(x)=\frac x2 + \frac{\cos x}{\sqrt x}.$$ Since $\lim\limits_{x\to\infty} \left(f(x)-\frac x2 \right)=0,$ the line $y=\frac x2$ is an asymptote of the graph of $f$ (see figure). However, they cut, even infinitely times.