Continuity at a point means that you can draw the function at that point without lifting up your pencil. Or in more technical terms, a small change in x produces only a small change in f(x)
In order for a function to be continuous at a point it must follow three criteria: 1) f(A) must be defined 2) f(x) has a limit as x approaches A from both direction 3) the limit as x approaches A is actually equal to f(A)
To visualize this, below is a continuous graph at point A. Notice how you could draw this without picking up your pencil.
Now below is a graph that is discontinuous for multiple reasons. Notice how you would have to pick up your pencil to draw this graph.