Does a tangent exist at $x=0$ to $y=sgn(x)$? Yesterday my professor told me that a tangent can be constructed at $x=0$ to the signum function reasoning that the two points considered while drawing a tangent must be close horizontally and not necessarily vertically.
He also told me that existence of tangent(s) isn't sufficient to prove that a derivative exists, i.e. a tangent can be drawn even when there exists no derivative.
Can anyone elaborate this point? Also, regarding the first statement, will this: $x=sgn(y)$ have a tangent at $x=0$? In this case, the two points now are close vertically and not horizontally...
Thanks in advance.

yes a vertical tangent can be constructed at x=0.
use first derivative principle to calculate left hand derivative and right hand derivative at x=0 , both will come infinity.Hence a vertical tangent .
