A Viscosity Subsolution or Supersolution of Class $C^2$ is Classical

Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$, let $\mathbb{R}$ be the set of real numbers, let $F$ be a second-order equation operator on $U$, and let $u:U\to\mathbb{R}$ be of class $C^2$ on $U$. No ellipticity hypothesis is imposed on $F$.

Then the following hold.

1. (Subsolutions) If $u$ is a viscosity subsolution of $F$ on $U$, then $u$ is a classical subsolution of $F$ on $U$.

2. (Supersolutions) If $u$ is a viscosity supersolution of $F$ on $U$, then $u$ is a classical supersolution of $F$ on $U$.