Finding a C1 Function for Continuous f,g in Real Numbers

[johnson12]

Hello,I need some advice on a problem.

Let $$f,g:R\rightarrow R$$ (where $$R$$ denotes the real numbers) be two continuous functions, assume that $$f(x) < g(x) \forall x \neq 0$$ ,

and f(0) = g(0).Define $$A = \left\{(x,y)\neq (0,0): y< f(x),x \in R\right\}$$

$$B = \left\{(x,y)\neq (0,0): y> g(x),x \in R\right\}$$Let $$r,s:R^{2}\rightarrow R$$ be $$C^{1}$$,and show that there is a $$C^{1}$$ function h defined on

$$R^{2}- \left\{(0,0)\right\}$$ such that h(x,y) = r(x,y) on A and h(x,y) = s(x,y) on B.

It seems that this follows from partitions of unity, but I am not sure where to
start, any suggestions at all are helpful.
Thanks.

 

[johnson12]

Hello,I need some advice on a problem.

Let $$f,g:R\rightarrow R$$ (where $$R$$ denotes the real numbers) be two continuous functions, assume that $$f(x) < g(x) \forall x \neq 0$$ ,

and f(0) = g(0).Define $$A = \left\{(x,y)\neq (0,0): y< f(x),x \in R\right\}$$

$$B = \left\{(x,y)\neq (0,0): y> g(x),x \in R\right\}$$


Let $$r,s:R^{2}\rightarrow R$$ be $$C^{1}$$,and show that there is a $$C^{1}$$ function h defined on

$$R^{2}- \left\{(0,0)\right\}$$ such that h(x,y) = r(x,y) on A and h(x,y) = s(x,y) on B.

It seems that this follows from partitions of unity, but I am not sure where to
start, any suggestions at all are helpful.
Thanks.

UPDATE: PROBLEM SOLVED