Let
In other words, the limit of
If
Equivalent Formulations of Continuous Functions
Let
Then the following four statements are logically equivalent:
- f is continuous at
- For every sequences and sequence
consisting of elements of with we have - For every
, there exists a such that for all with - 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
If
Similarly to real functions, we can have a sequential version of this definition:
Continuity is Preserved by Composition
see: function composition
If