Let be a subset of , and let be a function. Let be an element of . We say that the function is continuous at iff we have

In other words, the limit of as converges to in exists and is equal to .

If is continuous on every points on , we say it is continuous on .

Equivalent Formulations of Continuous Functions

Let be a subset of , let be a function, and let be an element of .

Then the following four statements are logically equivalent:

  1. f is continuous at
  2. For every sequences and sequence consisting of elements of with we have
  3. For every , there exists a such that for all with
  4. For every , there exists a such that for all with

The 3rd and 4th statements are the epsilon-delta definition of continuity.

Note

The difference between them is the usage of < and operators.

Tip

A useful consequence is that continuity can be very useful to calculate limits.

In Metric Space

Let be a metric space, and let be another metric space, and let be a function.

If we say that is continuous at iff for every there exists a such that whenever . We say that is continuous iff it is continuous at every point .

Similarly to real functions, we can have a sequential version of this definition: is continuous at iff whenever converges to with respect to the metric , then converges to with respect to metric

Continuity is Preserved by Composition

see: function composition If and are continuous at and respectively, so is at

See Also