Bounded

A function f from a set A ⊂ ℝ into ℝ is bounded if there exists M ∈ ℝ such that |f(x)| ≤ M for all x ∈ A.

Continuity at a point

A function f from a set A ⊂ ℝ into ℝ is continuous at a point p ∈ A if for every ε > 0 there exists δ > 0 such that |f(x) - f(p)| < ε for all x ∈ A with |x - p| < δ.

Uniform continuity

A function f from a set A ⊂ ℝ into ℝ is uniformly continuous on A for every ε > 0 there exists δ > 0 such that x, y ∈ A and |x - y| < δ imply |f(x) - f(p)| < ε.