To find a number $\delta$ such that:

$0 < |x - 3| < \delta \quad \text{implies} \quad |f(x) - 2| < 0.5,$

we need to analyze the behavior of the function $f(x)$ near $x = 3$ using the graph of $f$. The condition means that $f(x)$ is close to 2 when $x$ is close to 3.

Steps:

1. Find the range of $f(x)$ for $|f(x) - 2| < 0.5$: The inequality $|f(x) - 2| < 0.5$ can be rewritten as: $1.5 < f(x) < 2.5$ So, we are interested in values of $x$ where $f(x)$ stays between 1.5 and 2.5.

2. Find the corresponding $x$-interval around $x = 3$: Look at the graph of $f(x)$ and find where $f(x)$ stays within the range $(1.5, 2.5)$. Let's say the function stays within this range for values of $x$ between $x = 3 - \delta$ and $x = 3 + \delta$.

3. Determine $\delta$: From the graph, you need to estimate the value of $\delta$. This is the distance from $x = 3$ such that for all $x$ within this interval (i.e., $(3 - \delta, 3 + \delta)$), the value of $f(x)$ remains between 1.5 and 2.5.

If you provide the graph or describe its behavior near $x = 3$, I can help you find the exact value of $\delta$.