Let $f:\mathbb{R}^3 \to \mathbb{R}$ be a smooth function satisfying $\nabla f(\vec{p})\neq0$ for all $\vec{p} \in S$ where $S=\{\vec{x} \in \mathbb{R}^3\mid f(\vec{x})=0 \}$. I want to show that $S$ is a smooth orientable surface.
The definition I have for an orientable surface is that a surface $S$ is orientable if for every $\vec{p} \in S$ there exists a smooth choice of standard unit normal.
I know that since $\nabla f(\vec{p}) \neq0$ for all $\vec{p} \in S$ it follows from the Implicit Function Theorem that $S$ is indeed a smooth surface.
How do I check that $S$ is orientable though?