= Solution
The hypothesis makes $\Omega$ <star-shaped set>[star-shaped] with centre $z_0$. For small $h$ with the triangle having vertices $z_0,w,w+h$ contained in $\Omega$, the vanishing triangular integral gives
$$
g(w+h)-g(w)=\int_{[w,w+h]}f(z)\,dz.
$$
Parametrizing the final segment,
$$
\frac{g(w+h)-g(w)}h
=\int_0^1f(w+th)\,dt
\longrightarrow f(w)
$$
by continuity. Thus $g$ is <holomorphic function>[holomorphic] and
$$
\boxed{g'(w)=f(w)}.
$$
This is the triangle-integral construction behind <Morera's theorem>.
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