I'm currently showing that the following integral is continuous:
$$\int_{g_1(x)}^{g_2(x)} f(x,y) dy$$
Where $g_1, g_2, f$ are continuous.
I am doing this by taking the following limit:
$$\lim \limits_{(a,b) \to (x,y)} \int_{g_1(a)}^{g_2(a)} f(a, b) dy - \int_{g_1(x)}^{g_2(x)} f(x, y) dy$$
and showing that it $\to 0$.
My question is: for the first integral, should I replace $dy$ with a $db$? I think that I should but I'm not sure. Alternatively, is there a better way to express the limit so I don't run into this problem altogether?