what does it mean for a function like $f(t,z):[a,b] \times D \rightarrow \mathbb{C}$ to be continuous?

Question

I'm currently studying functions of one complex variable by John B.Conway and I met the following version of Leibniz's rule jhon.B.Conway,pg 68

But my problem is that throughout the book we haven't study functions like $\phi\ :[a,b] \times [c,d] \rightarrow \mathbb{C}$, what does it mean that this function is continuous? I also found a similar version of Leibniz's rule:

Let $f(t,z):[a,b] \times D \rightarrow \mathbb{C}$, $D \subseteq \mathbb{C}$ open, a continuous function analytic in $D$ for all $t \in [a,b]$. Also, $\frac{\partial f }{\partial z} (t,z) : [a,b] \times D \rightarrow \mathbb{C}$ is continuous. Then $g(z) = \int_a^b f(t,z) \, dz $ is analytic on $D$ with $$g'(z) = \int_a^b \frac{\partial f}{\partial z} (t,z) \, dt. $$ (P97, Complex Analysis, Frietag)

or:

Let $\Omega$ be an open set and let $\gamma$ be a rectifiable curve in $\mathbb{C}$. Suppose that $\varphi:\{\gamma\}\times\Omega\to\mathbb{C}$ (where $\{\gamma\}$ is the trace of $\gamma$) is a continuous function and define $g:\Omega\to\mathbb{C}$ by $$g(z)=\int_{\gamma}\varphi(w,z)\:dw.$$ Then $g(z)$ is continuous. If $\frac{\partial\varphi}{\partial z}$ exists for each $(w,z)$ in $\{\gamma\}\times\Omega$ and is continuous, then $g(z)$ is analytic and $$g'(z)=\int_{\gamma}\frac{\partial\varphi}{\partial z}(w,z)\:dw.$$

Similar to my first question, what does it mean that functions like $f(t,z):[a,b] \times D \rightarrow \mathbb{C}$ and $\varphi:\{\gamma\}\times\Omega\to\mathbb{C}$ being continuous? I've researching about this kind of functions and I think that continuous functions of this kind is related to the concept of Jointly continuous functions however I have not been able to find a definition for Join continuity without using concepts of topology (I have never study topology before). Moreover understanding this concept is really important because later in the book Conway introduced the concept of homotopic curves and he used this kind of functions whose domain is a Cartesian product . I'd be extremely grateful if someone could give a $\delta$ - $\epsilon$ definition or maybe a definition using sequences for this kind of continuous functions.

Another kind of problems when this functions arises is: enter image description here

related links:

• Continuity of cartesian product of functions between topological spaces
• Analogue of Leibniz's Rule
• Rigorous Proof of Leibniz's Rule for Complex
• Continuity and Joint Continuity
• How to prove xy is jointly continuous?

Answer 1

Jointly continuous for metric spaces. Let $A, B, C$ be metric spaces. A function $f : A \times B \to C$ is joinly continuous if: for any sequence $a_n$ in $A$ such that $a_n \to a$ and any sequence $b_n$ in $B$ such that $b_n \to b$, we have $f(a_n, b_n) \to f(a,b)$.

Answer 2

Sure... a function $f:[a,b]\times D\to \mathbb{C}$ is continuous at $(x,z)$ if for all $\varepsilon>0,$ there exists $\delta$ such that $|x-y|+|z-w|<\delta$ implies $|f(x,z)-f(y,w)|<\varepsilon$.