real analysis - Is this an equivalent definition of uniform continuity?

The motivation for this definition is that a main characteristic of functions defined on an interval which are continuous but not uniformly continuous is that the largest possible $\delta$ required to keep $\,f(x)\,$ within $\varepsilon$ of $\,f(x_0)\,$ can be made arbitrarily small by choosing the appropriate $x_0$. Consider, for example, $\,f(x) = \tan x\,$ in $\,\left(\dfrac{\pi}{2} - \varepsilon_0, \dfrac{\pi}{2}\right)$ for $\varepsilon_0$ small.
It is continuous everywhere in that interval but we are forced to choose smaller and smaller $\delta$ as $\,x_0 \to \dfrac{\pi}{2}$.

Let $\,f:I \to \mathbb{R}$ be a continuous real function. Consider some $\,x_0 \in I\,$ and fix $\,\varepsilon \in \mathbb{R_+}$. Define

$P = \left\lbrace\delta \mid \forall \,x \in I \ , \left\lvert x-x_0\right\rvert < \delta \implies \left\lvert \,f(x) - f(x_0)\right\rvert < \varepsilon\right\rbrace$.
Let $$\delta_{x_0}^* =
\begin{cases}
\sup P, & P \ \text{ bounded above} \\
1, & \text{otherwise}
\end{cases}
$$
We say $\,f$ is uniformly continuous if for all $\,\varepsilon \in \mathbb{R_+}, \:\inf\left\lbrace\delta_{x_0}^* \mid x_0 \in I \right\rbrace \in \mathbb{R_+}$.

It should be noted that the choice $1$ is arbitrary and $P$ is non empty by hypothesis. I'm not sure if this is equivalent, and if so how can one prove it is equivalent to the standard definition?