Loosely, a continuous function is one that has no "holes" (i.e., the function is undefined at a point), no "jumps" (i.e., the limit as x approaches a value c from the left is different from the limit as x approaches the same value c from the right), and no vertical asymptotes.

If you say that a function is continuous on an interval, then it has all of these properties on that interval. However, if the interval is open, then the function may have a vertical asymptote at one of the open endpoints and it will still be continuous.