Let $f(x)$ be an arbitrary function. Let $g(x) = \lfloor x\rfloor$ be the greatest integer function.
We know that $g(x)$ is discontinuous whenever $x$ is integer.
Can we say that $g(f(x)) = \lfloor f(x) \rfloor$ is discontinuous whenever $f(x)$ takes integer values?
Best Answer
No, we can't. If $f$ is constant then $g \circ f$ is constant and hence continuous.
By the way it's easy to characterize all functions $f:\mathbb R \to \mathbb R$ such that $g \circ f$ is continuous. We have $$g \circ f \quad \text{cont.}\quad \Leftrightarrow \quad g \circ f \quad \text{const.}\quad \Leftrightarrow \quad \exists k \in \mathbb Z, f(\mathbb R) \subseteq [k,k+1).$$