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I have been asked to tell in which points in $\mathbb R$ the limit does and does not exist.

$$\lim_{x\to x_0}\lfloor x\rfloor$$

Now, I have been thinking about first showing that the limit exists in all $x\in(\mathbb R / \mathbb Z)$

In order to do so, I tried to show that, for every $\epsilon>0$, I can find a $\delta>0$ so that the definition stands. I thought about choosing $\delta=\frac12\left(x-\lfloor x\rfloor\right)$ so that indeed I can always find a $\delta$ small enough to stay in my environment.

I'm having difficulties formalizing that. Also, I know that $\delta$ should not depend on $x$.

Where do I go from there?

Thanks!

Blue
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Yair Shemer
  • 41

1 Answers1

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Note that the function $x\to \lfloor x\rfloor=n$ is piecewise constant: if $x\in [n,n+1)$ with $n\in\mathbb{Z}$ then $\lfloor x\rfloor=n$.

Hence, if $x_0\not\in\mathbb{Z}$, for $\epsilon>0$, and $\delta:=\frac12\min\left(x_0-\lfloor x_0\rfloor,1+\lfloor x_0\rfloor-x_0\right)>0$, if $0<|x-x_0|<\delta$ then $|\lfloor x\rfloor-\lfloor x_0\rfloor|=0<\epsilon.$

Robert Z
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