Consider a Jordan curve $\Gamma\subset \mathbb{C}$ and let $\Omega$ be the interior of $\Gamma$ (which is well-defined by the Jordan curve theorem). Let $f:\Gamma\cup\Omega\rightarrow\mathbb{C}$ be analytic on $\Omega$ and continuous on $\Gamma$. Given these assumptions, does it hold that the limit
$$\lim_{w\rightarrow z}\frac{f(z)-f(w)}{z-w}$$
exists for each $z \in \Gamma$ (with $w \in \Gamma\cup\Omega$)? So basically I am asking if $f$ has a derivative on $\Gamma$ in an appropriately understood sense.