Continuous functions form a vector space

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 $\lambda \in \mathbb {R}$ and $f$ is a continuous function on $I$, then $\lambda f$ is also a continuous function on $I$.