Given a colloquial definition of uniform continuity as $f(x)$ and $f(y)$ can be made to be arbitrarily close when $x$ and $y$ are sufficiently close, and the distance between $x$ and $y$ is independent of $x$ and $y$.
I'm not really sure how to picture a uniformly continuous function in my head. I showed that if the derivative of a function is bounded, then it will be uniform continuous. (I had trouble with the converse though.) Thinking along the lines that I need to bound the change in $f$.
How do you visualize uniform continuity?
Best Answer
Fix $\varepsilon > 0$ and fix a $\delta > 0$ which works in the definition of uniform continuity.
The statement $|x-y| < \delta \Rightarrow |f(x)-f(y)| < \varepsilon$ tells you that you can place a rectangle of width $\delta$ and height $\varepsilon$ with its centre on any point on your graph, and the graph will always go through the middle of the rectangle, i.e. it never touches the top or bottom of the rectangle.