Give an example of a function that is uniformly continuous on R but is not bounded on R.
Question
Give an example of a function that is uniformly continuous on but is not bounded on .
Solution
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.
Similar Questions
Give an example of a bounded sequence in R that has no supremum, and prove that it has no supremum.
Give an example of a function f which satisfies the initial value problemI.V.P on a closed and bounded interval [a, b], but is not continuous on [a, b]
Show that the function f (x) := 1/x2 is uniformly continuous on (1, ∞), but that it isnot uniformly continuous on (0, ∞)
Give an example of a bounded non-empty set of real numbers which hasa supremum, but does not have a maximum. Justify your answercarefully.
Let f : R → R be continuous at c ∈ R. Then |f | is continuous at c. Give anexample to show that the reverse is not true
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.