Sum and Product Rules for One-Dimensional Derivatives and Continuity

Let $I$ be an interval, let $f,g:I\to\mathbb{R}$, and let $c$ be a real number. Here $f+g$, $cf$, and $fg$ denote the pointwise sum, scalar multiple, and product.

1. (Differentiability implies continuity) If $x_0\in I$ is an interior point of $I$ and $f$ is differentiable at $x_0$, then $f$ is continuous at $x_0$.

2. (Sum and constant multiple) If $x_0\in I$ is an interior point of $I$ and $f$ and $g$ are differentiable at $x_0$, then $f+g$ and $cf$ are differentiable at $x_0$ with

and every constant function on $I$ is differentiable at $x_0$ with derivative $0$.

3. (Product rule) If $x_0\in I$ is an interior point of $I$ and $f$ and $g$ are differentiable at $x_0$, then $fg$ is differentiable at $x_0$ with

4. (Continuity arithmetic) Let $E\subseteq\mathbb{R}$, let $\varphi,\psi:E\to\mathbb{R}$ be continuous at a point $x_0\in E$, and let $c\in\mathbb{R}$. Then $\varphi+\psi$, $c\varphi$, and $\varphi\psi$ are continuous at $x_0$, and every constant function on $E$ is continuous at $x_0$.