Lesson 1.3: Continuity

Closely associated with the concept of limits is the concept of continuity. Like limits, a graphical understanding of the concept is usually sufficient, though through the soon-to-be-studied techniques of delta-epsilon we can make the definition mathematically rigorous.

A function is termed continuous on an interval if, intuitively, small changes in its input result in small changes in its output. A function is continuous at a point if and only if all three of the following conditions are met:

1. is defined
2. exists (note that the limits from each side must be equal for this condition to be met)
3. 

We also say that a function is everywhere continuous, or simply continuous, if it is continuous at every point in its domain.

The necessary graphical understanding of continuity is fairly apparent from the three conditions above, but may be described as that the function has no holes or jumps on the specified interval.

Examples

• All polynomial functions are continuous.
  

  Graph of

• Rational functions are typically not continuous.
  

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• The sine and cosine functions are continuous…
  

  Graph of

• …but the other trigonometric functions are not.
  

  Graph of