Need help on understanding uniform continuity. The definition of uniform continuity is that for any $\epsilon > 0$, we can find specific fixed $\delta>0$ such that for any $x_1,x_2 \in D_f$ we have $|x_1-x_2|<\delta \to |f(x_1)-f(x_2)|<\epsilon$. For any $\epsilon$ we chose, the same $\delta$ must be able to make $|f(x_1)-f(x_2)|<\epsilon$ for any $x_1,x_2 \in D_f$. But I know there's a theorem that says any continuous function is uniformly continuous when its domain is closed. I have a hard time understanding this because a function that is very steep will make $\delta > 0$ to be changed.  

The function is becoming more steep as it gets closer to the right end point, $b$. And clearly, this is function is defined on the closed interval, $[0,b]$. But, we can still see that $\delta$ has to change at different points on the curve to make $|f(x_1)-f(x_2)|<\epsilon$. 
What am I missing? 
EDIT: http://www.math.uconn.edu/~kconrad/blurbs/analysis/metricspaces.pdf
I got my intuition from this paper, on page 30. It shows a graph where one function is continuous and other is uniformly continuous. 

Yes, $\delta$ must be smaller at the right end of your example. So given $\epsilon$, choose a $\delta$ that works at $b$. That $\delta$ will work everywhere.
