Is the floor of x continuous? Is $f(x)=[x]$, where $[x]$ denotes the floor function, continuous? 
Argument 1:
It seems to me that it is continuous, because each x value has a y value; the domain is all real numbers.
Argument 2:
However, I have heard other arguments saying that it is discontinuous at integers because:
Take the right hand limit: $$\lim_{x^+\to n}f(x)=n$$
Take the left hand limit: $$\lim_{x^-\to n}f(x-1)=n-1$$
Since these two are different, the function is discontinuous at integers. 
Which solution is correct? How do I internalize the second argument?
 A: Argument 1 does not speak to continuity at all.  It just says the function is well defined.  Any function that takes as input any real number and gives a unique value satisfies this.  
Argument 2 is correct, though the expression could be improved.  The definition of continuity is that $\lim_{x \to x_0}f(x)=f(x_0)$  Intuitively, this says that whenever $x$ is close to $x_0, f(x)$ is close to $f(x_0)$  This is not true for the floor function when $x_0$ is an integer.  There are values of $x$ very close to (and below) $x_0$ where the function values differ by $1$, not by a small number.
A: The way you want to write the second is 
$\lim_{x \rightarrow n^+} [x] = n$.  That is as $x$ approaches an integer from above.
And $\lim_{x \rightarrow n^-}[x] = n-1$
So it isn't continuous.

"It seems to me that it is continuous, because each x value has a y value; the domain is all real numbers."
That's not the definition of continuous.  That merely means that $[x]$ is a function.  Not all functions are continuous.
A: Argument 1 only says that the floor function is a function. There are lots of functions that aren't continuous, but to think about that you have to know the definition of "continuous" and internalize it.
I won't give the definition of continuous here, but can suggest a way to think about it. Whenever $x$ is strictly between $6$ and $7$ the floor of $x$ is $6$. The floor of $7$ is $7$. That means you can find values of $x$ as close to $7$ as you like without having the floor of $x$ close to the floor of $7$. That means the function is not continuous at $7$.
I think this is what you are trying to say your second argument, but
your notation in the second argument is confusing (even nonsense). Use words until you know how to use the symbols properly.
