Showing a function is continuous and analytic

Question

This is an exercise from Conway that I am stuck at. I think the function $φ$ is clearly continuous for points $(z, w)$ such that $z$ and $w$ are not equal. However I can't show that $φ$ converges to the derivative of $f$ at z as $(z, w)$ goes to points where $z$ and $w$ are equal... Also it is clear that for each fixed $w$, $f$ is analytic for points $z$ not equal to $w$. But how can I show that $f$ is analytic at $z=w$? Could anyone please help me with this problem?

Answer 1

Hint

$$\lim_{z \to w} \frac{f(z)-f(w)}{z-w}$$ is the definition of the derivative of $f$ at $w$.

Hint 2 Write the Taylor series of $f$ at $w$. Then, calculate from this $z \to \phi(z,w)$.