Wikipedia:Reference desk/Archives/Mathematics/2016 May 29

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May 29

Calculus:Bounded function,Uniform continuity,limits

Hi there,
A few questions: 1.I would like to know the limit of sin(x*D(x))/x at x=0. I believe it's defined and is 1, though I do suspect that it might be not defined. 2. Let f(x) be continuous, bounded, and monotonic between (0,1) (open interval). is it Uniform continued there? 3.if f is uniformly continuous at an interval, is f^2 is also is uniformly contentious there?

Thanks! — Preceding unsigned comment added by 87.70.75.110 (talk) 18:46, 29 May 2016 (UTC)

1. You didn't specify $D(x)$, but if it is sufficiently well-behaved (perhaps merely continuous is enough?) at 0, then the limit is its value there.
2. You don't even need monotonicity: if it is continuous, monotonic, and bounded, it has limits at the endpoints; the extension including them is a continuous function on a compact set and is uniformly continuous.
3. Yes: the interval is totally bounded, so the image is; then the square of that image is as well and then we can apply the previous result. --Tardis (talk) 19:07, 29 May 2016 (UTC)

1. D(x) is Nowhere continuous function.

Thanks! — Preceding unsigned comment added by 87.70.75.110 (talk) 19:21, 29 May 2016 (UTC)

For number 1, nothing can be said without conditions on D (you didn't even specify whether it approaches 0; however, a counterexample to that limit equaling 1 can be found by setting D to be the constant function 2; then the limit is 2, not 1). If D is nowhere continuous then you can say even less in generality, but the limit definitely need not equal 1 if it exists.

For number 2, continuity and boundedness are not enough to ensure uniform continuity, unless said continuity and boundedness are on a compact set. A counterexample is the function $\sin(1/x)$ defined on (0, 1). @Tardis: The counterexample also shows that a continuous and bounded function on an open interval need not have a limit at an endpoint. Monotonicity as an additional condition could be enough, but I haven't investigated thoroughly.

For number 3, yes. For any x and y in the domain of f we have $|f(x)^{2}-f(y)^{2}|=|f(x)+f(y)||f(x)-f(y)|$; let $\epsilon >0$ be given. Since a uniformly continuous function on a bounded interval is bounded, there is an A such that $|f(x)|<A$ for all x, and there is a $\delta$ such that $|x-y|<\delta \implies |f(x)-f(y)|<{\frac {\epsilon }{2A}}$; combining the inequalities shows that for any x and y, $|x-y|<\delta \implies |f(x)^{2}-f(y)^{2}|<\epsilon$, as desired.--Jasper Deng (talk) 05:02, 30 May 2016 (UTC)

So it does turn out that number 2 is true as-is. With monotonicity I'm sure that the function can then be continuously extended to the closure of the interval, and hence would then be uniformly continuous. So it's enough to show that it has a limit at both endpoints.

Theorem: Suppose f is a continuous, bounded, and monotonic function from the open interval (a, b) to the real numbers. Then f is uniformly continuous.

Proof: Suppose without loss of generality that f is increasing (the proof if it is decreasing is similar). Because f is bounded, and because the real numbers have the least upper bound property, we can set $L=\inf f((a,b)),U=\sup f((a,b))$. I now show that $\lim _{x\downarrow a}f(x)=L,\lim _{x\uparrow b}f(x)=U$; I prove the former, while the proof of the latter is similar. Suppose for contradiction that there existed an $\epsilon >0$ such that for all $\delta >0$, there is a y in the open interval $(a,a+\delta )$ such that $|f(y)-L|=f(y)-L\geq \epsilon$. Because f is monotonic, it follows that for every such y, then for every $z\geq y$ we have $f(z)\geq f(y)$. Now note that for any w in (a, b), choosing $\delta <w-a$ shows that $f(w)-L\geq \epsilon$. It follows that for all such w, $f(w)\geq L+\epsilon$. So $L+\epsilon$ is a lower bound on $f((a,b))$ (which, by the way, is the image of the interval (a,b) under f). But I said that $L=\inf f((a,b))$. So this is a contradiction. So it must instead hold that for every $\epsilon >0$ there is a $\delta >0$ such that for all x in $(a,a+\delta )$, we must have $f(x)-L<\epsilon$ - i.e. $\lim _{x\downarrow a}f(x)=L$, as desired.

Then the function g on [a, b] defined by $g(a)=L,g(b)=U,{\text{else}}\ g(x)=f(x)$ is continuous and bounded on the compact set [a, b], so by the Heine–Cantor theorem, g is uniformly continuous. Its restriction to (a, b) is evidently so as well. This completes the proof.--Jasper Deng (talk) 06:00, 30 May 2016 (UTC)

Thank you for the correction. I first thought of functions with unbounded derivative (like ${\sqrt {x}}$), but that's still uniformly continuous; thereafter I assumed that the function had to diverge (which makes the open interval restriction make sense). Convenient that the condition I ignored is what I wanted. --Tardis (talk) 15:07, 30 May 2016 (UTC)