Let (X, d) be a metric space. Which of the following is the correct definition for a function f: X → &mathbb{R} to be uniformly continuous on X?

Question

Mathematics · Real Analysis

Verified Solution4 OptionsUpdated: October 2026👁️1 View

Share:

Correct answer

For every ε > 0, there exists a δ > 0 such that for all x, y ∈ X, d(x, y) < δ implies |f(x) - f(y)| < ε.

Save this question and verified answer as a clean study sheet with live clickable link.

👥

Challenge your peer aspirants on WhatsApp or Telegram to solve this question!

Online Test Series

Practice full-length mock tests with real exam timers, instant score analysis, and all-India ranking.

Chapter Practice

Want more practice questions on Real Analysis?

Solve complete topic-wise tests with detailed solutions on GkGet.

Recommended practice questions in Mathematics · Real Analysis

Mahatma Gandhi's last fast unto death in 1948 was primarily against which of the following?

What was the primary meaning of 'Swaraj' as popularized by Mahatma Gandhi during India's independence movement?

What was one major impact of India’s independence on its economy?

Answer 1

For every ε > 0, there exists a δ > 0 such that for all x, y ∈ X, d(x, y) < δ implies |f(x) - f(y)| < ε.

Answer 2

f is uniformly continuous if and only if f is continuous on every open subset of X.

Answer 3

For every δ > 0, there exists an ε > 0 such that for all x, y ∈ X, |f(x) - f(y)| < ε implies d(x, y) < δ.

Answer 4

Uniform continuity of f always implies that f is a bounded function on X.

Related Practice Questions