Beta Function is Continuous and Positive on Positive Reals

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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 $\map \Beta {x, y} > 0$.



Sources