a) Let I be a bounded interval. Prove that if $f: I → \mathbb{R}$ is uniformly continuous on $I$, then f is bounded on $I$.
Proof: Suppose $I$ is a bounded interval, such that $I = (a,b)$. Then given $f: (a,b) → \mathbb{R}$, by theorem, $f$ is uniformly continuous on $(a,b)$ iff $f$ can be continuously extended to $[a,b]$; that is there is a continuous function $g: [a,b] → \mathbb{R}$ such that $f(x) = g(x)$ for $x$ in $(a,b)$.
Then since $f$ is uniformly continuous on $I$, there is a continuous function $g(x)$ such that $f(x) = g(x)$ on $[a,b]$.
Then $[a,b]$ is closed and bounded, so $g$ is bounded on $[a,b]$. Thus $f$ is also bounded on $I$.
b) Prove that part a) may be false if $I$ is unbounded or if $f$ is merely continuous.
Can someone please give a counterexample for part b).
Does my proof for part a) fine? Please can someone help me? Thanks.
$I=\mathbb{R}$ unbounded: $f(x)=x$ is uniformly continuous but unbounded.
$I=(0,1)$ bounded, $f(x)=1/x$ is continuous but not uniformly continuous.