We proved the following theorem in class and I have a few questions:
Theorem: Suppose that $f:[a,b]\times[c,d] \subset \mathbb{R}^2 \longrightarrow \mathbb{R}$ is a continuous function. Then the function
\begin{align} I:[c,d] &\longrightarrow\, \mathbb{R}\\ y\,\,\, &\longmapsto\, I(y) = \int_{a}^{b} f(x,y)\, dx \end{align}
is continuous on the interval $[c,d]$.
First of all we start to say that the function is uniform continuous over the domain. I know that a continuous function is uniform continuous if the domain is closed and bounded. How can see on the theorem that it is in fact closed?
Then we write the following ( Definition of uniform continuity ):
$ \forall \epsilon > 0 \exists \delta > 0 \Longrightarrow \mid \mid (h,k) \mid \mid < \delta \rightarrow \mid f(x+h,y+k) - f(x,y) \mid < \epsilon $
I know the definition of uniform continuity but I can't perfectly recognize it there. I guess the first part is the distance between h and k, but what are really those h and k?
Thanks in advance!