Viscosity solution for Hamilton-Jacobi equations and local extrema Consider a general first order nonlinear partial differential equation of the form (the so called Hamilton-Jacobi equation)
$$H(x,u,Du) = 0 \qquad x\in\Omega,\qquad[1]$$
where $\Omega$ is open in $\mathbb{R}^n$, $H:\Omega\times \mathbb{R}\times{R}^n\to \mathbb{R}$ is continuous and $u:\Omega\to \mathbb{R}$ is a function, $Du$ is the gradient of $u$. 
We say that a continuous function $u:\Omega\to\mathbb{R}$ is a viscosity subsolution of $[1]$ if for every $\phi\in C^1(\Omega)$ such that $u-\phi$ has a local minimum at $x\in \Omega$ one has $$H(x,u(x),D\phi(x)) \leq 0.$$
Similarly, we say that a continuous function $u:\Omega\to\mathbb{R}$ is a viscosity supersolution of $[1]$ if for every $\phi\in C^1(\Omega)$ such that $u-\phi$ has a local maximum at $x\in \Omega$ one has $$H(x,u(x),D\phi(x)) \geq 0.$$
I know that, for a fixed continuous function $u$, it is possible that for some $x \in \Omega$, there are no admissible test functions $\phi$ in the definition of viscosity sub- or supersolution. By the way it turns out the set of points at which there are admissible test functions is dense in $\Omega$. 
My question is the following: how to prove that, given $u$, there exist at least one $x \in \Omega$ for which we can find $\phi\in C^1(\Omega)$ such that $u-\phi$ has a local maximum (respect.maximum) at $x\in \Omega$?

Here is a short argument proving density of the set of points at which admissible test functions exist. The result requires some continuity of $u$ (semi-continuity is sufficient). 
Let $u$ be continuous and bounded and fix $x_0$. For $\varepsilon>0$ small let $x_\varepsilon$ be a point at which
$$u - \frac{1}{\varepsilon}|x-x_0|^2$$
attains its max (this max exists because $u$ is continuous and bounded; here, upper semi-continuity of $u$ is actually sufficient). So there is an admissible test function for the subsolution property at $x_\varepsilon$. Now show that as $\varepsilon\to 0$ we have $x_\varepsilon \to x_0$, which establishes density.
