At, $a$, the function has a "infinite slope" or vertical tangent line. If the slope of the tangent line is considered to be the instantaneous rate of change, at that point, the function increases "straight up".
Since the function increases "straight up", the next point would be right above the previous point. Since a function can only have one value in the range per domain, and this function would at least two range values (the points right above each other) for the same domain value, wouldn't it violate the definition of a function?
Note that I understand that the function doesn't actually consist of discrete points, but is instead continuous. However, if the function is considered to consist of points infinitesimally close together, wouldn't the next "infinitesimally" far apart point be right above the previous point?