Continuous functions form a vector space
Suppose is an interval (possibly open, closed, or infinite from the left side and possibly open, closed, or infinite from the right side -- so it could be of the form ). A continuous function on is a function on that is continuous at all points on the interior of and has the appropriate one-sided continuity at the boundary points (if they exist).
The continuous functions on form a real vector space, in the sense that the following hold:
- Additive: A sum of continuous functions is continuous: If are both continuous functions on , so is .
- Scalar multiplies: If and is a continuous function on , then is also a continuous function on .