Let $f(x):[a,b]\rightarrow\mathbb{R}$ be a $C^∞$-function. Show there exists a $C^∞$-function $F(x):\mathbb{R}\rightarrow\mathbb{R}$so that F(x)=f(x)

Question

Complete:Let $f(x):[a,b]\rightarrow\mathbb{R}$ be a $C^∞$-function with a,b in $\mathbb{R}$.Show that there exists a $C^∞$-function $F(x):\mathbb{R}\rightarrow\mathbb{R}$ so that F(x)=f(x) for every x in [a,b]. A given hint for this task is the theorem of Emile Borel:

For every sequence $(a_n)$ of real numbers there exists an indefinitely differentiable function f of compact support on $\mathbb{R}$ such that $f^{(n)}(0)=a_n$ for every n in $\mathbb{N}.$

However, I do not know how to come up with a suitable sequence $a_n$ so that I can use the Theorem and I would really appreciate some help.

Answer 1

Let $f_a\colon\mathbb{R}\longrightarrow\mathbb R$ be a $C^\infty$ function with compact support such that $(\forall n\in\mathbb{Z}_+):f_a^{(n)}(a)=f^{(n)}(a)$ and let $f_b\colon\mathbb{R}\longrightarrow\mathbb R$ be a $C^\infty$ function with compact support such that $(\forall n\in\mathbb{Z}_+):f_b^{(n)}(b)=f^{(n)}(b)$. Define$$F(x)=\begin{cases}f_a(x)&\text{ if }x<a\\f(x)&\text{ if }x\in[a,b]\\f_b(x)&\text{ if }x>b.\end{cases}$$