Good evening,
I'm currently working on this problem that involves proving that the following function is bijective:
$$f:[0,\alpha] \to [0,\beta], \space \alpha,\beta \in \mathbb{R}$$
$$f(\theta) = \frac{1}{2}\int_{0}^{\theta} g^2(s)ds$$
I've managed to prove that this function is actually injective by the definition. However, I'm finding it difficult to prove that this function is surjective. I've tried using the Mean Value Theorem to prove it, but I can't follow up with any conclusions.
To add up, if $g^2$ is non-zero in any point of the domain, can I conclude that $f$ will surely have an inverse?
I'm assuming $g$ can only reach values from $0$ to positive infinity, and $f(0) = 0, f(\alpha) = \beta $ (due to Darboux's theorem, I believe).
If possible, could you give me a clue on how to proceed with this? Thanks in advance!
$f(\theta)$ is a monotonically increasing function because $f'(\theta) = \frac{1}{2}g(\theta)^2 > 0$. Therefore, $f(\theta)$ is injective.
Since $f(\theta)$ is monotonically increasing function, $\min_\theta f(\theta) = f(0)$ and $\max_\theta f(\theta) = f(\alpha)$. And since $f(\theta)$ is continuous, its image is the interval $[f(0), f(\alpha)] = [0, \beta]$. Thus, $f(\theta)$ is also surjective, and therefore bijective.