%%visits: 5 If a function is closed bounded on an interval, and continuous, then it is [[uniform_continuity|uniformly_continuous]] ## intuition ## rigour If f ∈ C[a, b] then f is uniform continuous on [a, b]
Proof: By contradiction, assume ∃ϵ > 0 : ∀δ > 0, ∃x, x′ ∈ [a, b]: |x − x′| < δ, but, |f(x) − f(x′)| ≥ ϵ
Take $\delta = \frac{1}{n}: \exists x_n, x_{n'}: \left| x_n - x_n' \right| \le \frac{1}{n}$ but |f(xn) − f(xn′)| ≥ ϵ :todo: finish the proof. ## exam clinic ## examples and non-examples ## resources tags :math:real_analysis: