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 piece-wise 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?

Related differentiation topics

• Constant, Line, and Power Functions
• Exponential Functions
• Trigonometric Functions
• Constant Multiple
• Combinations: Sum, and Difference
• Combinations of Functions: Product and Quotient
• Composition of Functions (the Chain Rule)
• Transformations of Functions
• Inverses of Functions
• Hyperbolic Functions
• Linear Approximation
• Mean Value Theorem

Acknowledgements

Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.