Uniform Continuity and Limits

Question

Let $f\colon (0, 1]\to\mathbb{R}$ be a function which is uniformly continuous.

(i) Show that if $\{x_n\}$ is a sequence with $x_n > 0$ and $\lim_{n\to∞} x_n = 0$, then $\{f(x_n)\}$ is convergent.

(ii) Deduce from (i) that $\lim_{x→0} f(x)$ exists.

(iii) Is the function $f \colon (0, 1] → \mathbb{R}$, $f(x) = 1/x$ uniformly continuous?

Tried and got stuck. Anyone have an idea/outline of how to do this?

Answer 1

Hint: (i) With $f$ uniformly continuous -- for any $\epsilon > 0$ there is $\delta > 0$ such that $|x_m-x_n|< \delta $ implies $|f(x_m) - f(x_n)|< \epsilon$ .

Show that $(f(x_n))$ is a Cauchy sequence.

Hint: (iii) Assume $f(x) =1/x$ is uniformly continuous on $(0,1]$. Consider the sequence $(f(x_n))$ where $x_n \rightarrow 0,$ using the preceeding results.

Answer 2

In addition for ii), the limit will exist because of the definition of uniform continuity that limit as x->any p f(x) will exist.