Let $f:K \to \mathbb{R}$ be a $C^2$ function defined on the set $K=[0,1]^2$ (more generally, $K$ could be a convex and compact set in $\mathbb{R}^d$) such that $f$ and its derivatives are pointwise-bounded. Due to the latter assumption, $\nabla f$ is locally Lipschitz continuous with constant $L$, say in the $\ell_2$ norm. Notice that we are not assuming any convexity on $f$.
My question is:Can we extend $f$ and $\nabla f$ to $\mathbb{R}^2$ such that the gradient of the extension, call it $\nabla\mathcal{E}(f)$ still displays the same Lipschitz constant?
I think the answer is affirmative due to the regularity of $f$ (and the pointwise bounds) and would be essentially an extension of [1], but I am a bit unsure about how to justify it.