Question: Use the given graph of $f$ to find a number $\delta$ such that if $|x - 1| < \delta$ then $|f(x) - 1| < 0.2$.
1. **State the problem:** We want to find $\delta$ such that whenever $|x - 1| < \delta$, it follows that $|f(x) - 1| < 0.2$.
2. **Understand the meaning:** This means the values of $f(x)$ must stay within $0.2$ units of $1$, i.e., between $0.8$ and $1.2$.
3. **Use the graph information:** From the graph, we have:
- $f(0.7) = 1.2$
- $f(1) = 1$
- $f(1.1) = 0.8$
4. **Check the interval around $x=1$ where $f(x)$ stays within $0.8$ and $1.2$:**
- At $x=0.9$, $f(x)$ is just above $1.2$ (since $f(0.7)=1.2$ and the function is decreasing), so $0.9$ is close to the boundary.
- At $x=1.1$, $f(x) = 0.8$ which is the lower boundary.
5. **Determine $\delta$:** The interval around $1$ where $f(x)$ stays within $0.8$ and $1.2$ is approximately from $0.9$ to $1.1$.
6. **Calculate $\delta$:**
$$
\delta = \min(|1 - 0.9|, |1 - 1.1|) = \min(0.1, 0.1) = 0.1
$$
7. **Conclusion:** If $|x - 1| < 0.1$, then $|f(x) - 1| < 0.2$.
**Final answer:**
$$\boxed{\delta = 0.1}$$