If the derivatives of $f$ for $t\in L$ are real functions, then $f(\bar{z})=\overline{f(z)}, \forall z\in B(t,r).$

Question

Let $L\subset \mathbb{R}$ be an open interval and $f$ be a holomorphic function on $B(t,r)$ where $t\in L.$ How to prove the claim: If the derivatives of $f$ for $t\in L$ are real functions, then $$f(\bar{z})=\overline{f(z)}, \forall z\in B(t,r).$$

p.s. This is part of a proof of the Schwarz Reflection Principle.

Answer 1

A hint: Consider the Taylor expansion of $f$ centered at $t\in L$.

Another hint: Consider the function $g(z):=\overline{f(\bar z)}$ and check it for CR-ity.