$f,g \in C^\infty$ agree on a n-hood, yet are different?

Question

Can $f,g : \mathbb{R} \rightarrow \mathbb{R}$ be smooth functions that agree on a neighborhood, yet are not equal everywhere ?

Answer 1

As pointed out by @TheSilverDoe, $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=e^{−1/x^2}$ if $x>0$ and $f(x)=0$ if $x\leq0$, and $g: \mathbb{R} \rightarrow \mathbb{R}$ defined by $g(x)=0$ agree on $\mathbb{R}_{\leq0}$. Yet aren't equal on $\mathbb{R}_{>0}$.