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Find an example of a continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ with the following property,
For every $\epsilon >0$ there exists $\delta > 0$ such that $|f(x)-f(y)|< \epsilon\ $ whenever $x,y \in \mathbb{R}$ and $|x-y|< \delta$.
Now find an example of a continuous function for which the property doesn't hold.

Attempt;
From what I can gather , this question wants to functions, one that is continuous everywhere and when that is continuous but not everywhere continuous... is this correct?

Function for which the property holds
$f(x)=x$ is continuous everywhere and so the definition holds for all $x,y \in \mathbb{R}$

Function for which property doesn't hold

Are these correct, have I interpreted the question correctly?
Any help would be much appreciated.

otupygak
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1 Answers1

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As has already been mentioned by Martin and Benjamin, the exercise is about a property called uniform continuity. The only (and crucial) difference between continuity and uniform continuity is that the $\delta$ in the definition of the former may very well depend on the point $x$, while in the latter it is uniform over all possible $x$'s in the domain.

To be clearer, the definition of continuity states: for every point $x$, for every $\varepsilon>0$ there exist $\delta$ s.t. $\vert x- y\vert <\delta\Rightarrow \vert f(x) - f(y)\vert < \varepsilon$. As you can see, the first thing you specify in the definition is the point $x$, so everything else after that may depend on it.

On the other hand, your exercise implicitly gives the definition of uniform continuity, and there the specification of $x, y\in \mathbb R$ comes after you find $\delta$ such that the property holds. Thus, $\delta$ does not depend on the points you have chosen.

As you can see from previous comments, even very smooth functions like polynomials ($f(x) = x^2$) need not be uniformly continuous, so this is intuitively a very strong property.

To give another possible answer to your question, $f(x) = c$, with $c$ constant is trivially uniformly continuous, while for example $g(x) = e^x$ is continuous but not uniformly so.

Indigo
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