So I don't have a whole lot of experience in general proving continuity for multivariable functions, and I want to make sure I'm going about things correctly.
Prove that the function $B(z,w):=\int_0^1 t^{z-1}(1-t)^{w-1}dt$, for $z,w\in \mathbb{C}$, is continuous.
So I let $\varepsilon > 0$ and basically consider an expression of the form:
$|\int_0^1 t^{z-1}(1-t)^{w-1}dt - \int_0^1 t^{u-1}(1-t)^{v-1}dt| < \varepsilon$
Which simplifies into the form:
$|\int_0^1 t^{z-1}(1-t)^{w-1}[1-t^{u-z}(1-t)^{v-w}]dt| < \varepsilon$
And now to make this expression true I consider $|u-z|<\delta_1$ and $|v-w|<\delta_2$ and choose $\delta_1$ and $\delta_2$ as small as I need to (based on $\varepsilon$), which will allow me to make this integral as small as a wish, since $t^{u-z}(1-t)^{v-w}\rightarrow 1$ as $\delta_1,\delta_2 \rightarrow 0$.
Is this an acceptable way to go about continuity in several complex variables, should only a single delta equal to $Min[\delta_1,\delta_2]$ be used, are there any other subtleties I should be made aware of in the multivariable case?