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I was asked to prove/disprove the following:

If $f:\mathbb{R} \to \mathbb{R}$ continuous function, so for all $x \in \mathbb{R}$ and $\epsilon > 0$, exists $\delta > 0$ s.t. if $y \in \mathbb{R}$ and $|y-x|< \epsilon \to ֻֻ|f(y)-f(x)|< \delta$

So my intuition told me it was wrong for some reason (maybe because the requirement looks a bit like the definition of uniform continuity) so here is what I did:

Let's consider $f:\mathbb{R} \to \mathbb{R}$ defined as $f(x)=x^2$
So it's continuous:
Let $\epsilon >0$. We need to show that we can find $\delta_{x_0} >0$ such that $|x-x_0|<\delta_{x_0}$ implies $|f(x)-f(x_0)|<\epsilon$ $ \ \ $ $\forall x,x_0 \in \mathbb{R}$
We have $$|f(x)-f(x_0)|<|x^2-x_0^2|=|x+x_0||x-x_0|$$ And let's choose $0<\delta_{x_0}<1$ so $|x-x_0|<\delta_{x_0}$ implies $|x+x_0|<1+2|x_0|$ so: $$|f(x)-f(x_0)|<|x^2-x_0^2|=|x+x_0||x-x_0|<(1+2|x_0|)\delta_{x_0}$$ So we can choose $\delta_{x_0}<\frac{\epsilon}{1+2|x_0|}$ to get what we want.
Now I want to show it doesn't satisfy the claim:
Let $\epsilon >0$ and $x \in \mathbb{R}$ we need to try and find $\delta >0$ s.t. $|y-x|<\epsilon \to \ \ |f(y)-f(x)|<\delta$ . So:
$$|f(y)-f(x)|=|y+x||y-x|<(\epsilon-2|x|)\epsilon$$ But now I can just choose $\delta >(\epsilon-2|x|)\epsilon $ and the claim will hold.
So am I looking at this wrong? Or maybe the function I chose wasn't right?

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Notice that the property is true, just far weaker than continuity. The property hold if $f$ is locally bounded

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A function that is nowhere continuous and for which the given property holds is $$f(x)=\begin{cases}1&\text{if }x\in\mathbb Q\\0&\text{if }x\notin \mathbb Q\end{cases} $$

Given $x\in \mathbb R$ and $\epsilon>0$ you can pick $\delta=2$ and have that for all $y\in \mathbb R$ with (or without) $|x-y|<\epsilon$ we have $|f(x)-f(y)|<\delta$.

On the other hanbd, any $f\colon \mathbb R\to\mathbb R$ with your property is also continuous. This is because continujos $f$ is bounded on the compact interval $[x-\epsilon,x+\epsilon]$.

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  • $\begingroup$ Does this also proof that any continuous function is also satisfying the property mentioned? Plus, could you clarify a bit what's the difference between the property mentioned and the definition of continuity/uniform continuity please? $\endgroup$ – user258801 Aug 5 '15 at 14:53

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