Trouble understanding limits

[woven shoal]

A function $f$ is said to be continuous at a point $c$ if and only if
$$\lim_{x \to c} f(x) = f(c)$$

Based on this definition of continuity, the floor function $\lfloor x \rfloor$ is continuous at integer numbers, despite the fact that these are obviously discontinuities.

What am I getting wrong ?

I assume it is my calculation of the limit that causes the trouble. However, I don't understand how the following statement is false:
$$\lim_{x \to c} \lfloor x \rfloor = \lfloor c \rfloor$$
(where $c \in \mathbb{Z}$).

But if we replace $x$ with $c$, then yes, $\lfloor x \rfloor = \lfloor c \rfloor$. Thus the statement above should be true, right ?

Any help would be appreciated. Thanks in advance!