Question about the floor function.

Question

Does $f(\lfloor x \rfloor)=g(\lfloor x \rfloor)$ imply that $f(x)=g(x)$ ?

Answer 1

As long as $f(x)=g(x)$ for all $x \in \mathbb{Z}$ then you will have $f(\lfloor x \rfloor)=g(\lfloor x \rfloor)$. But the functions $f$ and $g$ can behave completely different on $\mathbb{R} \setminus \mathbb{Z}$. For example, $$ f(x) = \left \{ \begin{eqnarray*} x & , \quad x \in \mathbb{Z} \\ \pi &, \quad \mathbb{R} \setminus \mathbb{Z} \end{eqnarray*} \right. $$ and $$ g(x) = \left \{ \begin{eqnarray*} x & , \quad x \in \mathbb{Z} \\ \sqrt{2} &, \quad \mathbb{R} \setminus \mathbb{Z} \end{eqnarray*} \right. $$ have the property that $f(\lfloor x \rfloor)=g(\lfloor x \rfloor)$ but are clearly different functions.