# Beta Function is Continuous and Positive on Positive Reals

## 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

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 17.7 \ (4)$