If $f,g:\mathbb R \rightarrow \mathbb R$ are analytic and $f=g$ on an interval of positive length, can we conclude that $f=g$ everywhere?

Question

If $f,g:\mathbb R \rightarrow \mathbb R$ are analytic and $f=g$ on an interval of positive length, can we conclude that $f=g$ everywhere?

I guess it is more like a theorem than a problem.

I am thankful to get some for about it.

Answer 1

Hint: Suppose that $h: \mathbb{R} \to \mathbb{R}$ is analytic and on some open interval $J \subseteq \mathbb{R}$, it happens that $h|_{J} \equiv 0$. Can you conclude that $h \equiv 0$?

Imagine if $h \not\equiv 0$, then there is some $x \in \mathbb{R}$ such that $h(x) \neq 0$. If it happens that such an $x$ exists and $x > y$ for all $y \in J$, then let $x_0 = \inf\{(x > y)\forall y\in J \mid h(x) \neq 0\}$. Using the analytic behavior of $h$, conclude that there is some $\epsilon >0$ such that $h(x) = 0$ for all $x \in (x_0 - \epsilon, x_0 + \epsilon)$, a contradiction.