One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line

Let $\mathbb{R}$ be the real numbers, regarded also as the Euclidean space $\mathbb{R}^{1}$. Differentiability of a function on an interval at an interior point is that of Derivative at an Interior Point, and $f'(x)$ denotes the derivative there.

Then the following hold.

1. (The line is an interval with no boundary) $\mathbb{R}$ is an interval, and every $x\in\mathbb{R}$ is an interior point of it.

2. (Derivative and partial derivative agree) Let $f:\mathbb{R}\to\mathbb{R}$ and let $x\in\mathbb{R}$. Then $f$ is differentiable at $x$ if and only if the partial derivative of $f$ with respect to the first variable exists at $x$, and in that case $\partial_{1}f(x)=f'(x)$.

3. (Smoothness criterion) Let $D$ be a set of functions from $\mathbb{R}$ to $\mathbb{R}$ with the following property: every $f\in D$ is differentiable at every point of $\mathbb{R}$, and the function $\mathbb{R}\to\mathbb{R}$ sending $x$ to $f'(x)$ again belongs to $D$. Then every $f\in D$ is smooth on $\mathbb{R}^{1}$.