1. The problem statement, all variables and given/known data Give an example of a function f : R -> R where f is continuous and bounded but not uniformly continuous. 2. Relevant equations A function f : D -> R and R contains D, with Xo in D, and | X - Xo | < delta (X in D), implies | f(X) - f(Xo) | < epsilon. Then f is continuous at Xo. f is continuous if it is continuous at all Xo. f is uniformly continuous is close to continuity: " " x and y in D, | x - y | < delta then | f(x) - f(y) | < epsilon. 3. The attempt at a solution The problem I am having is creating a bounded function over the entirety of R, and then showing that the function is uniformly continuous. First I tried, f(x) = 0 when x = 0 x*sin(1/x) else Thinking that the craziness around 0 would make it not u.c. But this doesn't work because f is uniformly continuous, just not pretty. What I am thinking is that if a function that is continuous, then it is uniformly continuous on any closed interval for its domain. So, if I could find a function that is continuous but contains an open interval it would not be uniformly continuous. But as I say those words, I don't understand them. Any help that is great would be greatly appreciated. One more thing, the text I am using has not yet given a definition of a bounded function, but something very close, that if a function has a limit (or is continuous) at a point, then there is some bounded interval around that point.