Give an example of a function that is uniformly continuous on R but is not bounded on R.

Sure, an example of such a function is the identity function f(x) = x.

Step 1: Define the function Let's define the function f: R -> R by f(x) = x.

Step 2: Show that f is uniformly continuous on R A function f is uniformly continuous on a set if for every ε > 0, there exists a δ > 0 such that for all x, y in the set, if |x - y| < δ, then |f(x) - f(y)| < ε.

For the function f(x) = x, we can choose δ = ε. Then, for all x, y in R, if |x - y| < δ = ε, we have |f(x) - f(y)| = |x - y| < ε.

So, f is uniformly continuous on R.

Step 3: Show that f is not bounded on R A function f is bounded on a set if there exists a real number M such that |f(x)| ≤ M for all x in the set.

For the function f(x) = x, for any real number M, we can choose x = M + 1. Then, |f(x)| = |M + 1| > M.

So, f is not bounded on R.

Therefore, the function f(x) = x is an example of a function that is uniformly continuous on R but is not bounded on R.