Uniform continuity

As ever, analysis is turning my brain into an algebraic mush and preventing me from seeing sense.

I need to show that the function $f(x)=e^{-x^4}$ is uniformly continuous (or rather, I'm asked whether it is, but it clearly is so I just need to show it).

My attempts were too pathetic and embarrassing to warrant posting here, so I'd appreciate a poke in the right direction to get me started (nothing in the way of full answers though please).

On a side-note, I don't know what it is about uniform continuity that makes me instantly panic and run for the hills, but getting stumped on bog-standard easy questions like this certainly isn't helping! If anyone knows of any good books or websites on the matter, or indeed for analysis in general, then please let me know -- pure maths is brilliant and the only thing really keeping me from doing as well as I want to do in it is my fear of analysis.

Just a thought, could you use the MVT?

And here was I thinking you were infallible. :P I'll have a think, but first I'm told that Apostol's book on Analysis is v. good, probably worth a look.

(From a thread that I recycled, and then decided against recycling):

Bingo!

I can bound $|f'(x)|=|-4x^3e^{-x^4}|$ by 3, and so given $x,y \in \mathbb{R}$ there is some $z$ between $x$ and $y$ such that $|e^{-x^4}-e^{-y^4}| = |4z^3e^{-z^4}||x-y| \le 3|x-y|$, so given $\varepsilon > 0$, if $|x-y| < \frac{\varepsilon}{3}$ then $|f(x)-f(y)| < \varepsilon$.

Thanks for that


Thanks, I'll have a look into that! And I'm far from infallible; people here just think I'm infallible because I answer questions I know I can answer

Damn you, MVT! What chance does a 1st year who's only just learnt about it have of thinking to apply it in situations like these... :P

At the risk of exposing my ignorance here, doesn't the MVT only work on closed bounded intervals? In which case surely the proof doesn't extend to a proof over $\Re$? While we're on that topic, if we only want a proof working on closed bounded intervals, surely we have it anyway, since exp and -x^4 are both u. continuous on any closed bounded interval...

Unfortunately it's not quite as simple as composing functions, since $x^4$ isn't a uniformly continuous function on $\mathbb{R}$ (though it is continuous), and neither are $e^x$ or $e^{-x}$.

(From a thread that I recycled, and then decided against recycling):

Bingo!

I can bound $|f'(x)|=|-4x^3e^{-x^4}|$ by 3, and so given $x,y \in \mathbb{R}$ there is some $z$ between $x$ and $y$ such that $|e^{-x^4}-e^{-y^4}| = |4z^3e^{-z^4}||x-y| \le 3|x-y|$, so given $\varepsilon > 0$, if $|x-y| < \frac{\varepsilon}{3}$ then $|f(x)-f(y)| < \varepsilon$.

Thanks for that

I can bound $|f'(x)|=|-4x^3e^{-x^4}|$ by 3,

I much prefer it if people don't delete threads, if only because it provides a large set of questions for other people to practice with

I think the following is true and would immediately imply the result as well

Suppose f is continuous on the real line and f vanishes at +-infinity. Then f is uniformly continuous on the real line.

I am wondering how you got 3 as a bound. The easiest bound that I could see was 4.

I also think this is true (re post #7).

Infact, I think we can generalize the result to f(x) having limits at +-infinity. But now we're getting away from the original thread.

Yes, that's true; you can pick a closed interval $I$, on which it must be uniformly continuous, such that $|f(x)| < \varepsilon$ for all $x \not \in I$, and then the result follows pretty much immediately. Thanks for your help

A somewhat contrived method involving finding where the maximum of $f'(x)$ is (i.e. where $x=\frac{3}{4}$).

I don't know if I'm talking out my **** here but I generally find analysis easier if I try to do things like estimating rather than trying to find maxima/minima by differentiating. What I did was $|4x^3e^{-x^4}|\leq \frac{4|x|^3}{1+x^4}$ and then splitting into cases of |x|>1 and |x| less than or equal to 1.

Again, I'm also all for estimating rather than finding the actual maximum. (Excepting in those cases like contraction mapping where you may need the actual maximum).

So it's good that you mention estimation; it seems an obvious thing to use but I haven't given it much of my time to date, so I'll try and get good at it somehow.

On a side-note, I don't know what it is about uniform continuity that makes me instantly panic and run for the hills, but getting stumped on bog-standard easy questions like this certainly isn't helping!