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Let $x,y\in(a,b)$ be real numbers. I am trying to find a diffeomorphism $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying that $f(x)=y$ and $f(t)=t$ for all $t\notin(a,b)$. Here is my attempt.

Let $g\in C_0^\infty(\mathbb{R})$ such that ${\rm supp}(g)\subset(0,1)$ and $g(0)=1$. Define $g_s(t)=g(t/s)$. Now let $F(t)=t+(y-x)g_s(t-x)$, then we know $F(x)=y$ and $F$ is a diffeomorphism if $s$ is sufficiently large by the inverse function theorem. But the confusing point is that if $s$ is large, we can only "preserve" the value of $F$ outside a large interval. Hence I don't know how to do next...

Thanks for the help.

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