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I am struggling with this question:

Prove or give a counterexample: If $f$ is a continuous function on a compact subset $Y$ of a metric space $X$, then $f$ is uniformly continuous on $Y$.

Thanks for your help in advance.

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Basic real analysis should be a source of at least some intuition (which is misleading at times, granted). Can you think of some compact sets in $\mathbf R$? Are continuous functions on those sets uniformly continuous? Can you remember any theorems regarding those? Another idea is to start to try to prove the statement and see whether things start to fall apart. – Dylan Moreland Feb 18 '12 at 6:49
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Hint: Continuity tells you that for every $\epsilon\gt 0$ and every $x$, you can find a $\delta_x\gt 0$ such that $f(B(x,\delta_x))\subseteq B(f(x),\epsilon)$; for uniform continuity, you need a $\delta$ that does not depend on $x$. Now, if there were only finitely many values of $\delta_x$, then you could just pick the smallest one... – Arturo Magidin Feb 18 '12 at 7:19
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@ArturoMagidin Actually, just picking the smallest one is not good enough because then the two points might not lie in the same delta ball. – Rudy the Reindeer Feb 18 '12 at 8:12
    
@thecode: Do you know how to accept answers? – Rudy the Reindeer Feb 19 '12 at 10:03
up vote 43 down vote accepted

The answer is yes, if $f$ is continuous on a compact space then it is uniformly continuous:

Let $f: X \to Y$ be continuous, let $\varepsilon > 0$ and let $X$ be a compact metric space. Because $f$ is continuous, for every $x$ in $X$ you can find a $\delta_x$ such that $f(B(\delta_x, x)) \subset B({\varepsilon\over 2}, f(x))$. The balls $\{B(\delta_x, x)\}_{x \in X}$ form an open cover of $X$. So do the balls $\{B(\frac{\delta_x}{2}, x)\}_{x \in X}$. Since $X$ is compact you can find a finite subcover $\{B(\frac{\delta_{x_i}}{2}, x_i)\}_{i=1}^n$. (You will see in a second why we are choosing the radii to be half only.)

Now let $\delta_{x_i}' = {\delta_{x_i}\over 2}$.

You want to choose a distance $\delta$ such that for any two $x,y$ they lie in the same $B(\delta_{x_i}', x_i)$ if their distance is less than $\delta$.

How do you do that?

Note that now that you have finitely many $\delta_{x_i}'$ you can take the minimum over all of them: $\min_i \delta_{x_i}'$. Consider two points $x$ and $y$. Surely $x$ lies in one of the $B(\delta_{x_i}', x_i) $ since they cover the whole space and hence $x$ also lies in $B(\delta_{x_i}', x_i)$ for some $i$.

Now we want $y$ to also lie in $B(\delta_{x_i}', x_i)$. And this is where it comes in handy that we chose a subcover with radii divided by two:

If you pick $\delta : = \min_i \delta_{x_i}'$ (i.e. $\delta = \frac{\delta_{x_i}}{2}$ for some $i$) then $y$ will also lie in $B(\delta_{x_i}, x_i)$:

$d(x_i, y) \leq d(x_i, x) + d(x,y) < \frac{\delta_{x_i}}{2} + \min_k \delta_{x_k} \leq \frac{\delta_{x_i}}{2} + \frac{\delta_{x_i}}{2} = \delta_{x_i}$.

Hope this helps.

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The question being homework, isn't it better to give hints rather than a full answer? – wildildildlife Feb 18 '12 at 13:45
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Matt, I think you've missed another factor of 2. You have namely shown $y \in B(\delta_{x_i}, x_i)$, so that $d(x_{i},y) < \delta \implies d(f(x_{i}), f(y)) < \epsilon$, but you also need to show that $d(x,y) < \delta \implies d(f(x), f(y)) < \epsilon$. – Ryker Mar 12 '13 at 19:55
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@MattN., I think the original $\delta_{x_{i}}$ need to be chosen in such a way so as to satisfy the continuity condition for $\frac{\epsilon}{2}$ rather than $\epsilon$. Then you can use the triangle inequality in the final step after you've shown both x and y lie in the same ball. – Ryker Mar 12 '13 at 20:30
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@MattN.: Hmm, we chose $\delta_{x_{i}}$ in relation to a particular $x_{i}$, and the original condition is $d(x_{i},y)< \delta_{x_{i}} \implies d(f(x_{i}), f(y)) < \epsilon$. x, y in that ball are arbitrary and continuity of f at x would only imply we can find $\delta_{x}$, as well. But x is not one of the $x_{i}$ picked out by compactness, so $\delta$ doesn't necessarily relate to the $\delta_{x}$ needed. The one thing we do, however, know is that $d(x,y) \leq d(x, x_{i}) + d(x_{i}, y) < \epsilon + \epsilon = 2\epsilon$. – Ryker Mar 12 '13 at 23:41
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@RudytheReindeer Ryker is right, you are missing a factor of 2. You say in a comment that $\delta_{x_i}$ is such that $d(x,y)<\delta_{x_i}$ implies that $d(f(x),f(y))<\epsilon$. But that's not true. From continuity, $\delta_{x_i}$ is such that if both $x$ and $y$ are in the $B(\delta_{x_i},x_i)$, then $f(x)$ and $f(y)$ are both in the $B(\epsilon,f(x_i))$, which gives the distance between $f(x)$ and $f(y)$ at most $2\epsilon$, not $\epsilon$ . – Theo Mar 17 at 2:46

Let $(X, d)$ be a compact metric space, and $(Y, \rho)$ be a metric space. Suppose $f : X \to Y$ is continuous. We want to show that it is uniformly continuous.

Let $\epsilon > 0$. We want to find $\delta > 0$ such that $d(x,y) < \delta \implies \rho(f(x), f(y))< \epsilon$.

Ok, well since $f$ is continuous at each $x \in X$, then there is some $\delta_{x} > 0$ so that $f(B(x, \delta_{x})) \subseteq B(f(x), \frac{\epsilon}{2})$.

Now, $\{B(x, \frac{\delta_{x}}{2})\}_{x \in X}$ is an open cover of $X$, so there is a finite subcover $\{B(x_{i}, \frac{\delta_{x_{i}}}{2})\}_{i =1}^{n}$.

If we take $\delta := \min_{i} (\frac{\delta_{x_{i}}}{2})$, then we claim $d(x,y) < \delta \implies \rho(f(x), f(y)) < \epsilon$. Why?

Well, suppose $d(x,y) < \delta$. Since $x \in B(x_{i}, \frac{\delta_{x_{i}}}{2})$ for some $i$, we get $y \in B(x_{i}, \delta_{x_{i}})$. Why? $d(y, x_{i}) \leq d(y,x) + d(x,x_{i}) < \frac{\delta_{x_{i}}}{2} + \frac{\delta_{x_{i}}}{2} = \delta_{x_{i}}$.

Ok, finally, if $d(x,y) < \delta$, then we claim $\rho(f(x), f(y)) < \epsilon$. This is because $\rho(f(x), f(y)) \leq \rho(f(x), f(x_{i})) + \rho(f(x_{i}), f(y)) < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$.

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