If $f : [0,1) \rightarrow \mathbb{R}$ is uniformly continous, $f$ is bounded

Question

Prove or Disprove -

a) If $f:[0,1) \rightarrow \mathbb{R}$ is uniformly continuous in it's domain, $f$ is bounded.

b) If $f:[0,1] \rightarrow \mathbb{R}$ is uniformly continuous in it's domain, $f$ is bounded.

Attempt -

It feels like $b$ might be true and $a$ mustn't. Maybe some manipulation of the definition of uniformly continuity. I do want to say that because $f$ is bounded in both cases because we are talking about a bounded domain. But how should I proceed ? Thank you!

Answer 1

Any function which is uniformly continuous in an open bounded interval $(a,b)$ can be extended to a uniformly continuous function in $[a,b]$ (see for example Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous). Hence since any continuous function on a compact set is bounded, it follows that both (a) and (b) are true.

Answer 2

(b) Is because uniformly continuous implies continuous and this implies bounded if the domain is compact.

(a) Assume that it is unbounded. Take a sequence $x_n$ such that $f(x_n)\to\infty$. Without loss of generality $\to+\infty$. We must have $x_n\to 1$ because on compact subintervals of $[0,1)$ the function is bounded according to (b).

Therefore, $x_n$ gives a counterexample to uniform continuity. In fact, fixing say $\epsilon=1$ we can pick $n, M$ such that $|x_n-x_m|<\delta$ for all $m>M$ and then pick $m$ such that $f(x_m)>f(x_n)+2$. Therefore $|f(x_n)-f(x_m)|>\epsilon$. Since we are assuming that it is uniformly continuous, it follows that it can't be unbounded.

Answer 3

Both are true; the second one is trivially true since $[0,1]$ is compact and the continuous image of any compact set is still compact (in this case, since $[0,1]$ is closed and bounded, the image under $f$ will be closed and bounded).

The first one is true since $f$ can only vary by a bounded amount over a given $\Delta x$. Since $[0,1)$ is bounded, $f$ will have an upper bound.

Answer 4

A more general result that might help. I think this was an exercise in baby Rudin.

If $f: A \subset \mathbb{R} \rightarrow \mathbb{R}$ is uniformly continuous on $A$, and $A$ is bounded, then $f(A)$ is bounded.

One way to prove this is as follows: Given $\epsilon =1$, there exists some $\delta >0$ such that if $x, y \in A$, $$|x-y| < \delta \quad \Rightarrow \quad |f(x)-f(y)|<1$$ If $A$ is a bounded subset of $\mathbb{R}$, then there exists finitely many points $\{a_1,\dots,a_n\}$ in $A$ such that for every $x \in A$, there exists some $k =1,\dots,n$ such that $|x - a_k| < \delta$.

Let $M = \max \{|f(a_1)|,\dots,|f(a_n)|\}$. Then given any $x \in A$, $$|f(x)| \leq |f(a_k)| + 1 \leq M + 1$$