Question: Which of the following is/are true about a uniformly continuous function, $f$, on $[a, b]~$?
$a)~~$ The function is bounded.
$b)~~$ The function achieves its maximum on the set $(a, b)$.
$c)~~$ If $f(a) = 4$ and $f(b) = 6$ , then $f'() = 2$ for some $c ∈(,)$.
$d)~~$ The derivative $f'$ is bounded.
Solution:
- Part $(a)$ is true, because any continuous function on a closed interval is bounded.
- Part $(b)$ is also true of any continuous function on a closed interval.
- Part $(c)$ is not necessarily true, since the continuity doesn't necessarily imply differentiability.
- Part $(d)$ is true about uniformly continuous functions. The unboundedness of the derivative is what prevents finding the one size fits all $\delta$ for all $x$.
So here option $(a),~(b),~(c)$ are true. But given answer says only option $(a)$ is correct. Explanation given for Part $(b)$ and Part $(d)$ are as follows:
- Part $(b)$ is false because the maximum can be met at the end points.
- Part $(d)$ is false by the example, $~f(x)=x^{1/3}~,$ on $[-1,1]$ as it is uniformly continuous but $f'(x)$ is unbounded.
My question: From these two perspectives we have been getting two different answers. Anybody please help in this regard.
b) This is false. Suppose that $[a,b]=[-1,1]$ and take $f(x)=-x^2$. The maximum is attained at $-1$ and at $1$ and at no other point.
d) Take $f\colon[0,1]\longrightarrow\Bbb R$ defined by$$f(x)=\begin{cases}x^2\sin\left(\frac1{x^2}\right)&\text{ if }x>0\\0&\text{ otherwise.}\end{cases}$$