A function is continuous on an interval of its domain if it has no gaps, jumps, or vertical asymptotes in the interval. Another way to think informally about continuity is to ask: can I draw the graph of the function on the interval from one side to the other without picking up my pencil? Functions like lines and polynomials are continuous everywhere.
Try the following:
You can talk about continuity on an interval of the domain. For example, the functions with discontinuities above all are continuous on the interval -1 < x < 0, because that part of the domain misses the discontinuity.You can also ask whether a function is continuous, which (in most textbooks) is equivalent to asking whether the function is continuous on its domain. Note that this is a bit tricky, since y = 1/x is a continuous function by this definition, since the place where it has a discontinuity, at x = 0, is not in its domain. The fourth example in the applet, the jump discontinuity, is not a continuous function, because x = 1 is in its domain (as indicated by the filled-in point on the graph) and there is a discontinuity there.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.