Continuous functions form a unital algebra

Statement

Continuity at a point version

Suppose $c\in \mathbb{R}$. Then, the following are true:

• Additive closure: If $f,g$ are functions defined in open intervals containing $c$ and both of them are continuous at $c$, then the pointwise sum $f+g$ is continuous at $c$.
• Scalar multiplies: If $f$ is defined in an open interval containing $c$ and is continuous at $c$, and $\lambda$ is a real number, then $\lambda f$ is continuous at $c$.
• Multiplicative closure: If $f,g$ are functions defined in open intervals containing $c$ and both of them are continuous at $c$, then the pointwise product $f\cdot g$ is continuous at $c$.