We knew the $\varepsilon-\delta$ definition for continuous function: a function $f$ is continuous at $x_0$ iff for all $\varepsilon > 0$, there exists $\delta > 0$ such that $|f(x) - f(x_0)| < \varepsilon$ whenever $|x - x_0| < \delta$.
My question:
Let $f$ be continuous on some interval $[a, b]$ and $x_0 \in (a, b)$. With a given $\varepsilon$, how can we find $\delta$ such that $$|f(x) - f(x_0)| < \varepsilon \quad \text{ whenever } \quad |x - x_0| < \delta?$$
I think it is really difficult to have a "theoretical" answer, but I wonder if we have any numerical method to find $\delta$:
INPUT: function $f$, $x_0$ and $\varepsilon$.
OUTPUT: $\delta$.
Thank you very much.