Question

Is it possible that f(0)=0 and x=0 is the vertical asymptote of the function?

Answer 1

f(0)=0 is a point of origin, basically at x= 0, y= 0
x=0 can be a vertical asymptote of a function

Answer 2

f(0) = 0 means that x = 0 is part of the domain

but if you have x = 0 as the vertical asymptote, then x = 0 wouldn't be part of the domain.

So you have a contradiction.

Answer 3

What about a piece-wise function?
\[y= \frac{1}{x} ; (-\infty,0)\text{ and }(0,+\infty)\\y=x; \text{ for } x=0?\]

Answer 4

That's a good point, didn't think about that