The Difference Quotient of a Function with Bounded Continuous Derivative is Bounded, Symmetric and Continuous on the Plane

Adopt Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, and regard $\mathbb{R}$ as a metric space through the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Numbers. Every real number is an interior point of the interval $\mathbb{R}$ by Basic Facts about Intervals of the Real Line (i).

Let $\phi:\mathbb{R}\to\mathbb{R}$ be differentiable at every point of $\mathbb{R}$, let its derivative $\phi':\mathbb{R}\to\mathbb{R}$ be continuous as a map from $(\mathbb{R},d_{\mathbb{R}})$ to $(\mathbb{R},d_{\mathbb{R}})$, and let $L$ be a real number with $|\phi'(t)|\le L$ for every $t\in\mathbb{R}$. Define $F:\mathbb{R}^{2}\to\mathbb{R}$ by

$$F(x,y)=\frac{\phi(x)-\phi(y)}{x-y}\ \ \text{if }x\ne y,\qquad F(x,x)=\phi'(x).$$

1. (Bound and symmetry)(i) $|F(x,y)|\le L$ and $F(x,y)=F(y,x)$ for all $x,y\in\mathbb{R}$.

2. (Continuity)(ii) $F$ is continuous on $\mathbb{R}^{2}$.

3. (Measurability)(iii) $F$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{2})$ and $\mathcal{B}(\mathbb{R})$, the Borel $\sigma$-algebra of $\mathbb{R}^{2}$ and the Borel $\sigma$-algebra of the real line.