Yes―sort of. If displacement increases or decreases suddenly in an infinitely small point in time or a time interval that is too small to graph, a vertical line is used.
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infinite speed
No: not until instantaneous teleportation is discovered.
Object will change distance time graph when speed is changing. Distance time graph don't changed indicate of the stationary.
it will never be a vertical line as the slope is velocity and that would be infinite speed
No, because that would imply that the object travelled at infinite speed.