Similar question: For which $a$ does $\lim_{x\to a}f(x) = \lfloor x \rfloor$ exist?
The question is from Spivak's Calculus Chapter 5 Question 4. I am trying to look for the answer using the tools I have learned in this Chapter, i.e., the delta epsilon definition of the limit.
I attempted to prove the limit does not exist if $a$ is an integer, which I am unsure whether it is correct. And I am also having trouble proving the limit exists if $a$ is not an integer.
Assuming that $a$ is an integer.
If an interval $A$ containing integer $a$, we can find both $x_1$, and $x_2$ within the interval $A$, such that $f(x_1) = a$ and $f(x_2) = a-1$, for all $0 < |x_1 - a| < \delta_1$, and $0 < |x_2 - a| < \delta_2$. Let $\epsilon = \frac{1}{2}$. We need $|a-l|<\frac{1}{2}$, and $|a-1-l|<\frac{1}{2}$. And we cannot have that.
Is the logic correct above?
Assuming that $a$ is not an integer
How can I show that the limit exists using delta epsilon proof?
Let $a\in (n,n+1)$. Then there is a $\delta>0$ such that $(-\delta+a,a+\delta)\subset (n,n+1)$, and consequently, $f((-\delta+a,a+\delta)) = \{\lfloor a \rfloor\} = \{n\}$.