# Construction of a special diffeomorphism with some special properties

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.