Let $f\in C^\infty(0,\infty)$ and assume that $\lim_{x\to 0^+} f^{(k)}(x)=c_k$ exist and finite for each $k=0,1,2\cdots$. Show that $f$ can be extended to a $C^\infty$ function on $\mathbb R$.
One given hint is: define $g(x)=\sum_{k=0}^\infty c_k\frac{x^k}{k!}\chi(\epsilon_k^{-1}x)$ with a good smooth function $\chi$ and pick right $\{\epsilon_k\}$.
Take $\chi\in C^\infty_c([-1,1])$ and $\chi=1$ on $(-1/2,1/2)$. so we can pick $\epsilon_k$ to make $g$ is well-defined and let $g=f$ on $(0,\infty)$. And we can show that $g^{(k)}(0)=c_k$ for all $k$, but I have difficulty to show $\lim_{x\to 0^-}g^{(k)}(x)=c_k$.
To start this, I think one can fix $x\in(2^{-i},2^{-i+1})$, then the nontrivial part is $-1<\epsilon_k^{-1}x<- \frac{1}{2}$. Any idea to finish this end?