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Continuous functions form a vector space

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Statement

Suppose I 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,b],[a,b),(a,b),(a,b],(-\infty,b),(-\infty,b],(a,\infty),[a,infty),(-\infty,\infty)). A continuous function on I is a function on I that is continuous at all points on the interior of I and has the appropriate one-sided continuity at the boundary points (if they exist).

The continuous functions on I form a real vector space, in the sense that the following hold:

  • Additive: A sum of continuous functions is continuous: If f,g are both continuous functions on I, so is f + g.
  • Scalar multiplies: If \lambda \in \R and f is a continuous function on I, then \lambda f is also a continuous function on I.