Independently analytic and continuous, but not jointly continuous? In Bak/Newman's "Complex Analysis", they write:

17.9 Theorem
  Suppose $\phi(z,t)$ is a continuous function of $t$, with $b \ge t \ge a$, for fixed $z$ and an analytic function of $z \in D$ for fixed $t$. Then
  $$
f(z) = \int _a ^b \phi(z,t) \ dt
$$
  is analytic in $D$, and ...etc...

The proof starts off:

Since $f$ is a continuous function of $z$, according to Morera's Theorem we need only prove that ...etc...

I cannot seem to force $f$ to be continuous without requiring $\phi$ to be continuous in both variables together. I feel like it might be that analytic in the first variable and independently continuous in the second does not imply jointly continuous.
Question: What, if there is one, is an example which is analytic in the first, continuous in the second, but not jointly continuous? Any reasonable $\text{(domain) }D \times [a,b]$ is OK.
EDIT: Theorem 5.4 on p. 56 of Stein/Shakarchi's book here seems to be pretty much the same, except with the joint continuity assumption. (Their proof is neat, too, because it avoids blatantly using Fubini's theorem).
Thank you!
 A: It seems that you have to assume joint continuity, it does not follow from the assumptions. First, by Runge's theorem there exists a sequence of polynomials $p_n$ such that $p_n \to 0$ pointwise in the unit disk, but not uniformly in any neighborhood of $0$. For details see Pointwise convergence of sequences of holomorphic functions to holomorphic functions. Then you can define the function $\phi: \mathbb{D} \times [0,1] \to \mathbb{C}$ by $\phi(z,0) = 0$, $\phi(z,1/n) = p_n(z)$, and extend this piecewise linearly in $t$. The function $\phi$ is analytic in $z$ for every $t$, continuous in $t$ for every $z$, but it is not continuous at $(0,0)$. (If it were, then $p_n(z) = \phi(z,1/n)$ would converge uniformly to $0$ on every compact subset of the unit disk.)
Note that this construction does not guarantee that the function $f$ defined in the question is discontinuous, but I am pretty sure the Runge construction is flexible enough to adjust the construction in order to produce discontinuous $f$.
