Let $f:(0,1)→R$ be continuous. Pick out the statements which imply that $f$ is uniformly continuous.

Question

Let $f :(0, 1)→ R$ be continuous. Pick out the statements which imply that $f$ is uniformly continuous.

a. $|f(x) − f(y)| ≤ \sqrt{|x − y|}, \text{ for all }x, y \in [0, 1].$

b. $f\left(\frac{1}{n}\right)\rightarrow \frac{1}{2}$ and $f\left(\frac{1}{n^2}\right)\rightarrow \frac{1}{4}.$

Answer 1

For option $(b)$ does not implies that uniform convergence. Take $a_n=\frac{1}{n}$ and $b_n=\frac{1}{n^2}$.Here $a_n-b_n \rightarrow 0$ but $f(a_n)-f(b_n)\nrightarrow 0$ . Hence (b) does not implies uniform continuity.

Answer 2

Notice this

$$ |f(x) − f(y)| ≤ \sqrt{|x − y|} < \epsilon \implies |x-y|< \epsilon^2 = \delta. $$