Beta Function is Continuous and Positive on Positive Reals

Theorem
Let $x, y \in \R$ be real numbers.

Let $\map \Beta {x, y}$ be the Beta function:
 * $\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$

Let $y \in \R_{>0}$ be given.

Then $\map \Beta {x, y}$ is a positive and continuous function of $x$ on $\R_{>0}$.

Proof
For each $x > 0$, we have for all $t$ with $0 < t < 1$ that:


 * $t^{x - 1} \paren {1 - t}^{y - 1} > 0$

from which it is immediate that $\map \Beta {x, y} > 0$.