Calculus Applets

Continuity: Informal Approach

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:

1. The first graph shown, a simple parabola, is continuous everywhere. Move the slider and notice that the crosshair doesn't make any jumps. You could draw this curve without picking up your pencil.
   
   
2. Select the second example from the drop down menu. The sine curve has more wiggles in it, but it is still continuous.
   
   
3. Select the third example. This function has a vertical asymptote at x = 1. Move the slider and note that if you were drawing this curve, you'd have to pick up your pencil when you got to this point to move it to the other part of the curve. This is called an essential discontinuity.
   
   
4. Select the fourth example. This function jumps from 1 to 2 at x = 1, called a jump discontinuity. Moving the slider, it's clear you would also have to pick up your pencil at this point to draw the curve.
   
   
5. Select the fifth example. This function has a hole in it at x = 1, called a removable discontinuity. You don't have to pick up your pencil by much, but there still is a gap in the curve, even if it is only a single point. You can move the slider to exactly x = 1 by typing a 1 into the x = box to replace the value that's there. This will move the slider to that x value. What y value results in this case?

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.