Uniform continuity

Front

What is uniform continuity?

Back

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

Front

Are linear functions uniformly continuous on all of ℝ?

Back

Yes, linear functions are uniformly continuous on all of ℝ.

Front

Are quadratic functions uniformly continuous on all of ℝ?

Back

No, quadratic functions are not uniformly continuous on all of ℝ; they are only uniformly continuous on closed intervals.

Front

What does it mean for a function to be continuous?

Back

A function f is continuous at a point c if for every ε > 0, there exists a δ > 0 such that if |x - c| < δ, then |f(x) - f(c)| < ε.

Front

What is the relationship between uniform continuity and continuity?

Back

Uniform continuity implies continuity, but continuity does not imply uniform continuity.

Front

What is the continuous image of a compact set?

Back

The continuous image of a compact set is compact.

Front

What is the continuous image of a connected set?

Back

The continuous image of a connected set is connected.

Front

What is the continuous image of a convergent sequence?

Back

The continuous image of a convergent sequence converges.

Front

What is the continuous image of a Cauchy sequence?

Back

The continuous image of a Cauchy sequence is not necessarily Cauchy.

Front

What is the definition of a Cauchy sequence?

Back

A sequence {x_n} is Cauchy if for every ε > 0, there exists an N such that for all m, n > N, |x_m - x_n| < ε.