Some do and some don't. It's possible but not necessary.
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point
If the graph is a function, no line perpendicular to the X-axis can intersect the graph at more than one point.
x=0
It is impossible to determine the function here. (-1,4) is just a point on a graph.
If for every point on the horizontal axis, the graph has one and only one point corresponding to the vertical axis; then it represents a function. Functions can not have discontinuities along the horizontal axis. Functions must return unambiguous deterministic results.