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Let $f: [a,b]\times [c,d] \rightarrow \mathbb R$ be a smooth function of two variable (assuming that in boundary points $f$ has continuous one side partial derivatives).

Is a simple way to extend $f$ to a smooth function $F: \mathbb R \times \mathbb R \rightarrow \mathbb R$?

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up vote 2 down vote accepted

We can assume that $a=c=0$ and $b=d=1$. Then define $f_1(x,y)=f(-x,y)$ for $x\in [-1,0]$ and $y\in [0,1]$ (it will have continuous partial derivatives. Then define $f_2(x,-y)=f_1(x,y)$ for $y\in [-1,0]$ to extend the map to $[-1,1]\times [-1,1]$. Now we can repeat this procedure to extend the map to a smooth one on $\Bbb R^2$.

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