TITLE: How can this function be continuous at all points?
QUESTION [5 upvotes]: I have an exercise in which I need to determine which functions are continuous for all points
Note: $\lfloor x\rfloor$ is floor of $x$.
Function I need to consider is $f(x)= \lfloor x\rfloor+\sqrt{x- \lfloor x\rfloor}$
How can I prove that this function is continuous at integer values if neither  $\lfloor x\rfloor$ nor $\sqrt{x- \lfloor x\rfloor}$ are continuous at integer values?
I can determine it visually from graph and intuitively when considering values close to integers from both sides, but how can I rigorously prove it?

REPLY [7 votes]: It's simple to see that it suffices to prove the continuity of $f$ on the integer. So let $n\in\Bbb Z$ then:
$$\lim_{x\to n^+}f(x)=\lim_{x\to n^+}n+\sqrt{x-n}=n$$and$$\lim_{x\to n^-}f(x)=\lim_{x\to n^-}n-1+\sqrt{x-n+1}=n-1+1=n$$
so
$$\lim_{x\to n^+}f(x)=\lim_{x\to n^-}f(x)=n=f(n)$$
hence $f$ is continuous at $n$ and we conclude.