Can every $C^2$ function defined on a closed set in $\mathbb{R}^d$ be extended to $C^2(\mathbb{R}^d)$?

When reading Page 147 of the book "Continuous Martingales and Brownian Motion" by Daniel Revuz & Marc Yor, I am confused with the Remark $3^\circ$ of (3.3) Theorem (Ito's formula).In this remark, the author says:

"One gets another obvious extension when F is defined only on an open set but X takes a.s. its values in this set."

Moreover, in Page 341, Corollary 17.19(Local Ito's Formula) of the book "Foundations of Modern Probability" by Olav Kallenberg, he seems to use the extension of $C^2$ function defined on a closed subset of $\mathbb{R}^d$. But I am not sure what theorem he used. (Maybe Whitney's extension theorem (cf. ANALYTIC EXTENSIONS OF DIFFERENTIABLE FUNCTIONS DEFINED IN CLOSED SETS, Whitney, 1934.), but the paper is too difficult for me to read.)

My question:

1. Is there any extension theorem suitable for the situation here. Namely, can any function $f \in C^2(F)$, $F \subset \mathbb{R}^d$ is a closed subset, be extended to a function in $C^2(\mathbb{R}^d)$? Then I can use the method in the book by Kallenberg, to prove this so called "local Ito's formula".
2. Or is there any method to prove this Ito's formula?

Thanks a lot!