# real analysis – Looking for a reference for an extension problem of function

Let $$Omega_1$$, $$Omega_2$$ be bounded,convex, open domains with smooth boundary in $$mathbb{R}^2$$ and $$overlineOmega_1subsetOmega_2$$. Suppose we are given a $$C^1$$ function $$f:overlineOmega_1cup(mathbb{R}^2setminusOmega_2)rightarrowmathbb{R}$$ satisfies the following properties:

(1)$$fequiv 1$$ on $$partialOmega_1$$ and $$fequiv 0$$ on $$partialOmega_2$$.

(2)$$nabla fcdot nu_k< 0$$ on $$partialOmega_k$$, where $$nu_k$$ is the unit outward normal vector to $$Omega_k$$, ($$k=1,2$$).

Now the question is that can we extend $$f$$ to $$Omega_2setminusoverlineOmega_1$$ such that $$0leq fleq 1$$ and $$|nabla f|neq 0$$ in $$Omega_2setminusoverlineOmega_1$$?

I think the answer must be yes because I can imagine its figure as a frustum of a cone. I have read some references about the Whitney’s extension theorem but they do not match. I would be very appreciate if anyone can provide the proof or references.

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