A Limit Computed from the Epsilon-Delta Definition

In the setting of The Real Line: Standing Notation and Background for Calculus, in which $\mathbb{R}$ is recorded as an interval containing at least two points, let $f:\mathbb{R}\to\mathbb{R}$ be given by

which we also write $x^2$.

Problem. Show, directly from Limit of a Real Function at a Point of an Interval and without appealing to any theorem about limits or about continuity, that

That is: for each real $\varepsilon>0$, exhibit a real $\delta>0$ such that every $x\in\mathbb{R}$ satisfying $0<|x-3|<\delta$ satisfies $|x^2-9|<\varepsilon$.