Beta Function is Continuous and Positive on Positive Reals

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Theorem

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

Let $\Beta \left({x, y}\right)$ be the Beta function:

$\displaystyle \Beta \left({x, y}\right) := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \left({1 - t}\right)^{y - 1} \ \mathrm d t$


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

Then $\Beta \left({x, y}\right)$ 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:

$t^{x-1} \left({1-t}\right)^{y-1} > 0$

from which it is immediate that $\Beta \left({x, y}\right) > 0$.



Sources