Continuity: An informal discussion
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.
1. A continuous function
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. A sine curve
Select the second example from the drop down menu.
The sine curve has more wiggles in it, but it is still
continuous.
3. Essential discontinuity
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. A jump discontinuity
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.
Jump discontinuities often occur with piecewise defined functions.
5. A removable discontinuity
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?
Other differentiation topics
Acknowledgements
Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.
